Loogle!

Result

Found 266 declarations mentioning Submodule.subtype. Of these, only the first 200 are shown.

• Submodule.subtype 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : ↥p →ₗ[R] M
• LinearMap.subtype_comp_codRestrict 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ (b : M), f b ∈ p) : p.subtype ∘ₛₗ LinearMap.codRestrict p f h = f
• Submodule.injective_subtype 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Function.Injective ⇑p.subtype
• Submodule.subtype_injective 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : Function.Injective ⇑p.subtype
• Submodule.coe_subtype 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) : ⇑p.subtype = Subtype.val
• Submodule.subtype_apply 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} (x : ↥p) : p.subtype x = ↑x
• Submodule.subtype_comp_inclusion 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (p q : Submodule R M) (h : p ≤ q) : q.subtype ∘ₗ Submodule.inclusion h = p.subtype
• LinearMap.subtype_comp_restrict 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} {f : M →ₛₗ[σ₁₂] M₂} {p : Submodule R M} {q : Submodule R₂ M₂} (hf : ∀ x ∈ p, f x ∈ q) : q.subtype ∘ₛₗ f.restrict hf = f.domRestrict p
• Module.End.submodule_pow_eq_zero_of_pow_eq_zero 📋 Mathlib.Algebra.Module.Submodule.LinearMap
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {g : Module.End R ↥N} {G : Module.End R M} (h : G ∘ₗ N.subtype = N.subtype ∘ₗ g) {k : ℕ} (hG : G ^ k = 0) : g ^ k = 0
• Submodule.map_comap_subtype 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (p p' : Submodule R M) : Submodule.map p.subtype (Submodule.comap p.subtype p') = p ⊓ p'
• LinearMap.comap_domRestrict 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} (p : Submodule R₂ M₂) (f : M₂ →ₛₗ[σ₂₁] M) (p' : Submodule R M) : Submodule.comap (f.domRestrict p) p' = Submodule.comap p.subtype (Submodule.comap f p')
• Submodule.disjoint_iff_comap_eq_bot 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : Disjoint p q ↔ Submodule.comap p.subtype q = ⊥
• LinearMap.map_domRestrict 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} [RingHomSurjective σ₂₁] (p : Submodule R₂ M₂) (f : M₂ →ₛₗ[σ₂₁] M) (p' : Submodule R₂ ↥p) : Submodule.map (f.domRestrict p) p' = Submodule.map f (Submodule.map p.subtype p')
• LinearMap.comap_codRestrict 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (hf : ∀ (c : M₂), f c ∈ p) (p' : Submodule R ↥p) : Submodule.comap (LinearMap.codRestrict p f hf) p' = Submodule.comap f (Submodule.map p.subtype p')
• LinearMap.map_codRestrict 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} [RingHomSurjective σ₂₁] (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (h : ∀ (c : M₂), f c ∈ p) (p' : Submodule R₂ M₂) : Submodule.map (LinearMap.codRestrict p f h) p' = Submodule.comap p.subtype (Submodule.map f p')
• LinearMap.comap_restrict 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} {p : Submodule R₂ M₂} {q : Submodule R M} {f : M₂ →ₛₗ[σ₂₁] M} (h : ∀ x ∈ p, f x ∈ q) (p' : Submodule R ↥q) : Submodule.comap (f.restrict h) p' = Submodule.comap p.subtype (Submodule.comap f (Submodule.map q.subtype p'))
• LinearMap.map_restrict 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₂₁ : R₂ →+* R} [RingHomSurjective σ₂₁] {p : Submodule R₂ M₂} {q : Submodule R M} {f : M₂ →ₛₗ[σ₂₁] M} (h : ∀ x ∈ p, f x ∈ q) (p' : Submodule R₂ ↥p) : Submodule.map (f.restrict h) p' = Submodule.comap q.subtype (Submodule.map f (Submodule.map p.subtype p'))
• Submodule.comapSubtypeEquivOfLe 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (hpq : p ≤ q) : ↥(Submodule.comap q.subtype p) ≃ₗ[R] ↥p
• Submodule.comapSubtypeEquivOfLe_apply_coe 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (hpq : p ≤ q) (x : ↥(Submodule.comap q.subtype p)) : ↑((Submodule.comapSubtypeEquivOfLe hpq) x) = ↑↑x
• Submodule.comapSubtypeEquivOfLe_symm_apply 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (hpq : p ≤ q) (x : ↥p) : (Submodule.comapSubtypeEquivOfLe hpq).symm x = ⟨⟨↑x, ⋯⟩, ⋯⟩
• Submodule.ker_subtype 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : p.subtype.ker = ⊥
• LinearMap.ker_domRestrict 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (p : Submodule R M) (f : M →ₛₗ[τ₁₂] M₂) : (f.domRestrict p).ker = Submodule.comap p.subtype f.ker
• LinearMap.ker_restrict 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {p : Submodule R M} {q : Submodule R₂ M₂} {f : M →ₛₗ[τ₁₂] M₂} (hf : ∀ x ∈ p, f x ∈ q) : (f.restrict hf).ker = Submodule.comap p.subtype f.ker
• LinearMap.ker_submoduleMap 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {τ₂₁ : R₂ →+* R} [RingHomInvPair τ₁₂ τ₂₁] (f : M →ₛₗ[τ₁₂] M₂) (p : Submodule R M) : (f.submoduleMap p).ker = Submodule.comap p.subtype f.ker
• LinearMap.le_ker_iff_comp_subtype_eq_zero 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {N : Submodule R M} {f : M →ₛₗ[τ₁₂] M₂} : N ≤ f.ker ↔ f ∘ₛₗ N.subtype = 0
• LinearMap.exists_ne_zero_of_sSup_eq_top 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} {f : M →ₛₗ[τ₁₂] M₂} (h : f ≠ 0) (s : Set (Submodule R M)) (hs : sSup s = ⊤) : ∃ m ∈ s, f ∘ₛₗ m.subtype ≠ 0
• LinearMap.comp_ker_subtype 📋 Mathlib.Algebra.Module.Submodule.Ker
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (f : M →ₛₗ[τ₁₂] M₂) : f ∘ₛₗ f.ker.subtype = 0
• Submodule.range_subtype 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : p.subtype.range = p
• Submodule.map_subtype_le 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (p' : Submodule R ↥p) : Submodule.map p.subtype p' ≤ p
• Submodule.map_subtype_top 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : Submodule.map p.subtype ⊤ = p
• LinearMap.subtype_comp_rangeRestrict 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) : f.range.subtype ∘ₛₗ f.rangeRestrict = f
• Submodule.comap_subtype_self 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : Submodule.comap p.subtype p = ⊤
• Submodule.map_subtype_range_inclusion 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {p p' : Submodule R M} (h : p ≤ p') : Submodule.map p'.subtype (Submodule.inclusion h).range = p
• Submodule.comap_subtype_eq_top 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {p p' : Submodule R M} : Submodule.comap p.subtype p' = ⊤ ↔ p ≤ p'
• Submodule.range_inclusion 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p q : Submodule R M) (h : p ≤ q) : (Submodule.inclusion h).range = Submodule.comap q.subtype p
• LinearMap.range_codRestrict 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₂₁ : R₂ →+* R} [RingHomSurjective τ₂₁] (p : Submodule R M) (f : M₂ →ₛₗ[τ₂₁] M) (hf : ∀ (c : M₂), f c ∈ p) : (LinearMap.codRestrict p f hf).range = Submodule.comap p.subtype f.range
• LinearMap.range_restrict 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂] (f : M →ₛₗ[τ₁₂] M₂) {p : Submodule R M} {q : Submodule R₂ M₂} (h : ∀ x ∈ p, f x ∈ q) : (f.restrict h).range = Submodule.comap q.subtype (Submodule.map f p)
• Submodule.comap_subtype_le_iff 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] {p q r : Submodule R M} : Submodule.comap p.subtype q ≤ Submodule.comap p.subtype r ↔ p ⊓ q ≤ p ⊓ r
• Submodule.map_subtype_embedding_eq 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (p' : Submodule R ↥p) : (Submodule.MapSubtype.orderEmbedding p) p' = Submodule.map p.subtype p'
• Submodule.coe_mapIic_apply 📋 Mathlib.Algebra.Module.Submodule.Range
  {R : Type u_1} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (q : Submodule R ↥p) : ↑(p.mapIic q) = Submodule.map p.subtype q
• LinearEquiv.toLinearMap_ofTop 📋 Mathlib.Algebra.Module.Submodule.Equiv
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {h : p = ⊤} : ↑(LinearEquiv.ofTop p h) = p.subtype
• Submodule.equivSubtypeMap 📋 Mathlib.Algebra.Module.Submodule.Equiv
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (q : Submodule R ↥p) : ↥q ≃ₗ[R] ↥(Submodule.map p.subtype q)
• Submodule.comap_equiv_self_of_inj_of_le_apply 📋 Mathlib.Algebra.Module.Submodule.Equiv
  {R : Type u_1} {M : Type u_4} {N : Type u_8} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {f : M →ₗ[R] N} {p : Submodule R N} (hf : Function.Injective ⇑f) (h : p ≤ f.range) (x : ↥(Submodule.comap f p)) : (Submodule.comap_equiv_self_of_inj_of_le hf h) x = (LinearMap.codRestrict p (f ∘ₗ (Submodule.comap f p).subtype) ⋯) x
• Submodule.equivSubtypeMap_symm_apply 📋 Mathlib.Algebra.Module.Submodule.Equiv
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R ↥p} (x : ↥(Submodule.map p.subtype q)) : ↑↑((p.equivSubtypeMap q).symm x) = ↑x
• Submodule.equivSubtypeMap_apply 📋 Mathlib.Algebra.Module.Submodule.Equiv
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} {q : Submodule R ↥p} (x : ↥q) : ↑((p.equivSubtypeMap q) x) = (p.subtype.domRestrict q) x
• Submodule.map_subtype_span_singleton 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} (x : ↥p) : Submodule.map p.subtype (R ∙ x) = R ∙ ↑x
• Submodule.isCompl_comap_subtype_of_isCompl_of_le 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_10} {M : Type u_11} [Ring R] [AddCommGroup M] [Module R M] {p q r : Submodule R M} (h₁ : IsCompl q r) (h₂ : q ≤ p) : IsCompl (Submodule.comap p.subtype q) (Submodule.comap p.subtype r)
• Submodule.biSup_comap_subtype_eq_top 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_9} (s : Set ι) (p : ι → Submodule R M) : ⨆ i ∈ s, Submodule.comap (⨆ i ∈ s, p i).subtype (p i) = ⊤
• Finsupp.restrictDom_comp_subtype 📋 Mathlib.LinearAlgebra.Finsupp.Supported
  {α : Type u_1} {M : Type u_2} {R : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set α) [DecidablePred fun x => x ∈ s] : Finsupp.restrictDom M R s ∘ₗ (Finsupp.supported M R s).subtype = LinearMap.id
• Finsupp.linearCombination_restrict 📋 Mathlib.LinearAlgebra.Finsupp.LinearCombination
  {α : Type u_1} {M : Type u_2} (R : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] {v : α → M} (s : Set α) : Finsupp.linearCombination R (s.domRestrict v) = (Submodule.span R (v '' s)).subtype ∘ₗ Finsupp.linearCombinationOn α M R v s ∘ₗ ↑(Finsupp.supportedEquivFinsupp s).symm
• LinearIndependent.linearCombination_comp_repr 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
  {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (hv : LinearIndependent R v) : Finsupp.linearCombination R v ∘ₗ hv.repr = (Submodule.span R (Set.range v)).subtype
• Submodule.finrank_map_subtype_eq 📋 Mathlib.LinearAlgebra.Dimension.Finrank
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (q : Submodule R ↥p) : Module.finrank R ↥(Submodule.map p.subtype q) = Module.finrank R ↥q
• Submodule.sup_eq_range 📋 Mathlib.LinearAlgebra.Prod
  {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p q : Submodule R M) : p ⊔ q = (p.subtype.coprod q.subtype).range
• Submodule.exists_eq_graph 📋 Mathlib.LinearAlgebra.Prod
  {S : Type u_4} {H : Type u_6} {I : Type u_7} [Semiring S] [AddCommMonoid H] [Module S H] [AddCommMonoid I] [Module S I] {G : Submodule S (H × I)} (hf₁ : Function.Bijective (Prod.fst ∘ ⇑G.subtype)) : ∃ f, G = f.graph
• Submodule.exists_equiv_eq_graph 📋 Mathlib.LinearAlgebra.Prod
  {S : Type u_4} {H : Type u_6} {I : Type u_7} [Semiring S] [AddCommMonoid H] [Module S H] [AddCommMonoid I] [Module S I] {G : Submodule S (H × I)} (hG₁ : Function.Bijective (Prod.fst ∘ ⇑G.subtype)) (hG₂ : Function.Bijective (Prod.snd ∘ ⇑G.subtype)) : ∃ e, G = (↑e).graph
• LinearMap.kerComplementEquivRange_symm_apply 📋 Mathlib.LinearAlgebra.Prod
  {R : Type u_3} {M : Type u_4} {M₂ : Type u_5} [Ring R] [AddCommGroup M] [AddCommGroup M₂] [Module R M] [Module R M₂] (f : M →ₗ[R] M₂) {C : Submodule R M} (h : IsCompl C f.ker) (a✝ : ↥f.range) : (f.kerComplementEquivRange h).symm a✝ = (LinearEquiv.ofInjective (LinearMap.codRestrict f.range (f ∘ₗ C.subtype) ⋯) ⋯).toEquiv.symm ((LinearEquiv.ofTop (LinearMap.codRestrict f.range (f ∘ₗ C.subtype) ⋯).range ⋯).toEquiv.symm a✝)
• Submodule.strictMono_comap_prod_map 📋 Mathlib.LinearAlgebra.Quotient.Basic
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p : Submodule R M) : StrictMono fun m => (Submodule.comap p.subtype m, Submodule.map p.mkQ m)
• LinearMap.ker_le_range_iff 📋 Mathlib.LinearAlgebra.Quotient.Basic
  {R : Type u_1} {M : Type u_2} {R₂ : Type u_3} {M₂ : Type u_4} {R₃ : Type u_5} {M₃ : Type u_6} [Ring R] [Ring R₂] [Ring R₃] [AddCommMonoid M] [AddCommGroup M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {τ₁₂ : R →+* R₂} {τ₂₃ : R₂ →+* R₃} [RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₃] M₃} : g.ker ≤ f.range ↔ f.range.mkQ ∘ₗ g.ker.subtype = 0
• Submodule.dualRestrict_def 📋 Mathlib.LinearAlgebra.Dual.Defs
  {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (W : Submodule R M) : W.dualRestrict = W.subtype.dualMap
• iSupIndep.linearEquiv_apply 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [DecidableEq ι] [Ring R] [AddCommGroup N] [Module R N] {p : ι → Submodule R N} (ind : iSupIndep p) (iSup_top : ⨆ i, p i = ⊤) (x : Π₀ (i : ι), ↥(p i)) : (ind.linearEquiv iSup_top) x = (DFinsupp.sumAddHom fun i => (p i).subtype.toAddMonoidHom) x
• Submodule.iSup_eq_range_dfinsupp_lsum 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [Semiring R] [AddCommMonoid N] [Module R N] [DecidableEq ι] (p : ι → Submodule R N) : iSup p = ((DFinsupp.lsum ℕ) fun i => (p i).subtype).range
• Submodule.biSup_eq_range_dfinsupp_lsum 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [Semiring R] [AddCommMonoid N] [Module R N] [DecidableEq ι] (p : ι → Prop) [DecidablePred p] (S : ι → Submodule R N) : ⨆ i, ⨆ (_ : p i), S i = (((DFinsupp.lsum ℕ) fun i => (S i).subtype) ∘ₗ DFinsupp.filterLinearMap R (fun i => ↥(S i)) p).range
• Submodule.mem_iSup_iff_exists_dfinsupp 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [Semiring R] [AddCommMonoid N] [Module R N] [DecidableEq ι] (p : ι → Submodule R N) (x : N) : x ∈ iSup p ↔ ∃ f, ((DFinsupp.lsum ℕ) fun i => (p i).subtype) f = x
• lsum_comp_mapRange_toSpanSingleton 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [DecidableEq ι] [Semiring R] [AddCommMonoid N] [Module R N] [(m : R) → Decidable (m ≠ 0)] (p : ι → Submodule R N) {v : ι → N} (hv : ∀ (i : ι), v i ∈ p i) : ((DFinsupp.lsum ℕ) fun i => (p i).subtype) ∘ₗ (DFinsupp.mapRange.linearMap fun i => LinearMap.toSpanSingleton R ↥(p i) ⟨v i, ⋯⟩) ∘ₗ ↑(finsuppLequivDFinsupp R) = Finsupp.linearCombination R v
• iSupIndep_of_dfinsupp_lsum_injective 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [DecidableEq ι] [Semiring R] [AddCommMonoid N] [Module R N] (p : ι → Submodule R N) (h : Function.Injective ⇑((DFinsupp.lsum ℕ) fun i => (p i).subtype)) : iSupIndep p
• Submodule.mem_biSup_iff_exists_dfinsupp 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [Semiring R] [AddCommMonoid N] [Module R N] [DecidableEq ι] (p : ι → Prop) [DecidablePred p] (S : ι → Submodule R N) (x : N) : x ∈ ⨆ i, ⨆ (_ : p i), S i ↔ ∃ f, ((DFinsupp.lsum ℕ) fun i => (S i).subtype) (DFinsupp.filter p f) = x
• iSupIndep_iff_forall_dfinsupp 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [DecidableEq ι] [Semiring R] [AddCommMonoid N] [Module R N] (p : ι → Submodule R N) : iSupIndep p ↔ ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((DFinsupp.lsum ℕ) fun i => (p i).subtype) (DFinsupp.erase i v) = ↑x → x = 0
• iSupIndep.dfinsupp_lsum_injective 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [DecidableEq ι] [Ring R] [AddCommGroup N] [Module R N] {p : ι → Submodule R N} (h : iSupIndep p) : Function.Injective ⇑((DFinsupp.lsum ℕ) fun i => (p i).subtype)
• iSupIndep_iff_dfinsupp_lsum_injective 📋 Mathlib.LinearAlgebra.DFinsupp
  {ι : Type u_1} {R : Type u_3} {N : Type u_6} [DecidableEq ι] [Ring R] [AddCommGroup N] [Module R N] (p : ι → Submodule R N) : iSupIndep p ↔ Function.Injective ⇑((DFinsupp.lsum ℕ) fun i => (p i).subtype)
• LinearMap.subToSupQuotient 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : ↥p →ₗ[R] ↥(p ⊔ p') ⧸ Submodule.comap (p ⊔ p').subtype p'
• LinearMap.comap_leq_ker_subToSupQuotient 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : Submodule.comap p.subtype (p ⊓ p') ≤ (LinearMap.subToSupQuotient p p').ker
• LinearMap.quotientInfToSupQuotient 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : ↥p ⧸ Submodule.comap p.subtype p ⊓ Submodule.comap p.subtype p' →ₗ[R] ↥(p ⊔ p') ⧸ Submodule.comap (p ⊔ p').subtype p'
• LinearMap.quotientInfEquivSupQuotient 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : (↥p ⧸ Submodule.comap p.subtype p ⊓ Submodule.comap p.subtype p') ≃ₗ[R] ↥(p ⊔ p') ⧸ Submodule.comap (p ⊔ p').subtype p'
• LinearMap.quotientInfEquivSupQuotient_injective 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : Function.Injective ⇑(LinearMap.quotientInfToSupQuotient p p')
• LinearMap.quotientInfEquivSupQuotient_surjective 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : Function.Surjective ⇑(LinearMap.quotientInfToSupQuotient p p')
• LinearMap.quotientInfEquivSupQuotient_apply_mk 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) (x : ↥p) : have map := Submodule.inclusion ⋯; (LinearMap.quotientInfEquivSupQuotient p p') (Submodule.Quotient.mk x) = Submodule.Quotient.mk (map x)
• LinearMap.quotientInfEquivSupQuotient_symm_apply_left 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) (x : ↥(p ⊔ p')) (hx : ↑x ∈ p) : (LinearMap.quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = Submodule.Quotient.mk ⟨↑x, hx⟩
• LinearMap.quotientInfEquivSupQuotient_symm_apply_right 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) {x : ↥(p ⊔ p')} (hx : ↑x ∈ p') : (LinearMap.quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0
• LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] {p p' : Submodule R M} {x : ↥(p ⊔ p')} : (LinearMap.quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0 ↔ ↑x ∈ p'
• LinearMap.coe_quotientInfToSupQuotient 📋 Mathlib.LinearAlgebra.Isomorphisms
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] (p p' : Submodule R M) : ⇑(LinearMap.quotientInfToSupQuotient p p') = ⇑(LinearMap.quotientInfEquivSupQuotient p p')
• Module.End.invtSubmodule.map_subtype_mem_of_mem_invtSubmodule 📋 Mathlib.Algebra.Module.Submodule.Invariant
  {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (f : Module.End R M) {p : Submodule R M} (hp : p ∈ f.invtSubmodule) {q : Submodule R ↥p} (hq : q ∈ Module.End.invtSubmodule (LinearMap.restrict f hp)) : Submodule.map p.subtype q ∈ f.invtSubmodule
• LinearMap.IsProj.subtype_comp_codRestrict 📋 Mathlib.LinearAlgebra.Projection
  {S : Type u_5} [Semiring S] {M : Type u_6} [AddCommMonoid M] [Module S M] {U : Submodule S M} {f : M →ₗ[S] M} (hf : LinearMap.IsProj U f) : U.subtype ∘ₗ hf.codRestrict = f
• LinearMap.surjective_comp_subtype_of_isComplemented 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p : Submodule R E} {M : Type u_6} [AddCommMonoid M] (h : IsComplemented p) [Module R M] : Function.Surjective fun f => f ∘ₗ p.subtype
• LinearMap.ofIsCompl_subtype_zero_eq 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p q : Submodule R E} (hpq : IsCompl p q) : LinearMap.ofIsCompl hpq p.subtype 0 = p.projection q hpq
• Submodule.linearProjOfIsCompl_comp_subtype 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p q : Submodule R E} (h : IsCompl p q) : p.projectionOnto q h ∘ₗ p.subtype = LinearMap.id
• Submodule.projectionOnto_comp_subtype 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p q : Submodule R E} (h : IsCompl p q) : p.projectionOnto q h ∘ₗ p.subtype = LinearMap.id
• Submodule.subtype_comp_projectionOnto_add_eq_id 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p q : Submodule R E} (h : IsCompl p q) : p.subtype ∘ₗ p.projectionOnto q h + q.subtype ∘ₗ q.projectionOnto p ⋯ = LinearMap.id
• Submodule.toLinearMap_symm_quotientEquivOfIsCompl 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p q : Submodule R E} (h : IsCompl p q) : ↑(p.quotientEquivOfIsCompl q h).symm = p.mkQ ∘ₗ q.subtype
• LinearMap.ker_id_sub_eq_of_proj 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p : Submodule R E} {f : E →ₗ[R] ↥p} (hf : ∀ (x : ↥p), f ↑x = x) : (LinearMap.id - p.subtype ∘ₗ f).ker = p
• LinearMap.ofIsCompl_eq' 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {p q : Submodule R E} (h : IsCompl p q) {φ : ↥p →ₗ[R] F} {ψ : ↥q →ₗ[R] F} {χ : E →ₗ[R] F} (hφ : φ = χ ∘ₗ p.subtype) (hψ : ψ = χ ∘ₗ q.subtype) : LinearMap.ofIsCompl h φ ψ = χ
• Submodule.projectionOnto_comp_subtype_eq_zero 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {p q : Submodule R E} (h : IsCompl p q) : p.projectionOnto q h ∘ₗ q.subtype = 0
• Submodule.coe_prodEquivOfIsCompl 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] (p q : Submodule R E) (h : IsCompl p q) : ↑(p.prodEquivOfIsCompl q h) = p.subtype.coprod q.subtype
• Submodule.isIdempotentElemEquiv_symm_apply_coe 📋 Mathlib.LinearAlgebra.Projection
  {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] (p : Submodule R E) (f : { f // ∀ (x : ↥p), f ↑x = x }) : ↑(p.isIdempotentElemEquiv.symm f) = p.subtype ∘ₗ ↑f
• LinearMap.exact_subtype_mkQ 📋 Mathlib.Algebra.Exact.Basic
  {R : Type u_8} {N : Type u_10} [Ring R] [AddCommGroup N] [Module R N] (Q : Submodule R N) : Function.Exact ⇑Q.subtype ⇑Q.mkQ
• LinearMap.exact_subtype_ker_map 📋 Mathlib.Algebra.Exact.Basic
  {R : Type u_8} {N : Type u_10} {P : Type u_11} [Ring R] [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (g : N →ₗ[R] P) : Function.Exact ⇑g.ker.subtype ⇑g
• Function.Exact.linearMap_rangeRestrict 📋 Mathlib.Algebra.Exact.Basic
  {R : Type u_1} {M : Type u_2} {N : Type u_4} {P : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} : Function.Exact ⇑f ⇑g → Function.Exact ⇑f.range.subtype ⇑g.rangeRestrict
• Function.Exact.iff_linearMap_rangeRestrict 📋 Mathlib.Algebra.Exact.Basic
  {R : Type u_1} {M : Type u_2} {N : Type u_4} {P : Type u_6} [Semiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} : Function.Exact ⇑f ⇑g ↔ Function.Exact ⇑f.range.subtype ⇑g.rangeRestrict
• Submodule.set_smul_eq_map 📋 Mathlib.Algebra.Module.Submodule.Finsupp
  {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (sR : Set R) (N : Submodule R M) [SMulCommClass R R ↥N] : sR • N = Submodule.map (N.subtype ∘ₗ (Finsupp.lsum R) (DistribSMul.toLinearMap R ↥N)) (Finsupp.supported (↥N) R sR)
• Subalgebra.toSubmodule_subtype 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
  {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) : (Subalgebra.toSubmodule S).subtype = ↑S.val
• DirectSum.isInternal_biSup_submodule_of_iSupIndep 📋 Mathlib.Algebra.DirectSum.Module
  {R : Type u} [Ring R] {ι : Type v} [dec_ι : DecidableEq ι] {M : Type u_1} [AddCommGroup M] [Module R M] {A : ι → Submodule R M} (s : Set ι) (h : iSupIndep fun i => A ↑i) : DirectSum.IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)
• skewAdjointPart_comp_subtype_skewAdjoint 📋 Mathlib.Algebra.Star.Module
  (R : Type u_1) {A : Type u_2} [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] : skewAdjointPart R ∘ₗ (skewAdjoint.submodule R A).subtype = LinearMap.id
• selfAdjointPart_comp_subtype_selfAdjoint 📋 Mathlib.Algebra.Star.Module
  (R : Type u_1) {A : Type u_2} [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] : selfAdjointPart R ∘ₗ (selfAdjoint.submodule R A).subtype = LinearMap.id
• skewAdjointPart_comp_subtype_selfAdjoint 📋 Mathlib.Algebra.Star.Module
  (R : Type u_1) {A : Type u_2} [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] : skewAdjointPart R ∘ₗ (selfAdjoint.submodule R A).subtype = 0
• selfAdjointPart_comp_subtype_skewAdjoint 📋 Mathlib.Algebra.Star.Module
  (R : Type u_1) {A : Type u_2} [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] : selfAdjointPart R ∘ₗ (skewAdjoint.submodule R A).subtype = 0
• StarModule.decomposeProdAdjoint_symm_apply 📋 Mathlib.Algebra.Star.Module
  (R : Type u_1) (A : Type u_2) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A] [Module R A] [StarAddMonoid A] [StarModule R A] [Invertible 2] (a : ↥(selfAdjoint A) × ↥(skewAdjoint A)) : (StarModule.decomposeProdAdjoint R A).symm a = (selfAdjoint.submodule R A).subtype a.1 + (skewAdjoint.submodule R A).subtype a.2
• Submodule.smithNormalFormOfLE 📋 Mathlib.LinearAlgebra.FreeModule.PID
  {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] [IsDomain R] [IsPrincipalIdealRing R] [Finite ι] (b : Module.Basis ι R M) (N O : Submodule R M) (N_le_O : N ≤ O) : (o : ℕ) × (n : ℕ) × Module.Basis.SmithNormalForm (Submodule.comap O.subtype N) (Fin o) n
• Module.Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord 📋 Mathlib.LinearAlgebra.FreeModule.PID
  {ι : Type u_1} {R : Type u_2} [CommRing R] {M : Type u_3} [AddCommGroup M] [Module R M] {n : ℕ} {N : Submodule R M} (snf : Module.Basis.SmithNormalForm N ι n) {i : Fin n} : snf.bM.coord (snf.f i) ∘ₗ N.subtype = snf.a i • snf.bN.coord i
• Submodule.map_range_rTensor_subtype_lid 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {Q : Type u_2} [CommSemiring R] [AddCommMonoid Q] [Module R Q] {I : Submodule R R} : Submodule.map (↑(TensorProduct.lid R Q)) (LinearMap.rTensor Q I.subtype).range = I • ⊤
• Submodule.exists_fg_le_subset_range_rTensor_subtype 📋 Mathlib.RingTheory.TensorProduct.Finite
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (s : Set (TensorProduct R N M)) (hs : s.Finite) : ∃ J, ∃ (_ : J.FG), s ⊆ ↑(LinearMap.rTensor M J.subtype).range
• Submodule.exists_fg_le_eq_rTensor_subtype 📋 Mathlib.RingTheory.TensorProduct.Finite
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R N M) : ∃ J, ∃ (_ : J.FG), ∃ y, x = (LinearMap.rTensor M J.subtype) y
• LinearPMap.coe_vadd 📋 Mathlib.LinearAlgebra.LinearPMap
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] {σ : R →+* S} {E : Type u_4} [AddCommGroup E] [Module R E] {F : Type u_5} [AddCommGroup F] [Module S F] (f : E →ₛₗ[σ] F) (g : E →ₛₗ.[σ] F) : ↑(f +ᵥ g) = ⇑(f ∘ₛₗ g.domain.subtype) + ↑g
• LinearMap.exists_extend 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
  {K : Type u_3} {V : Type u_4} {V' : Type u_5} [DivisionRing K] [AddCommGroup V] [AddCommGroup V'] [Module K V] [Module K V'] {p : Submodule K V} (f : ↥p →ₗ[K] V') : ∃ g, g ∘ₗ p.subtype = f
• LinearMap.exists_extend_of_notMem 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
  {K : Type u_3} {V : Type u_4} {V' : Type u_5} [DivisionRing K] [AddCommGroup V] [AddCommGroup V'] [Module K V] [Module K V'] {p : Submodule K V} {v : V} (f : ↥p →ₗ[K] V') (hv : v ∉ p) (y : V') : ∃ g, g ∘ₗ p.subtype = f ∧ g v = y
• Module.fgSystem.equiv_comp_of 📋 Mathlib.Algebra.Colimit.Finiteness
  {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] [DecidableEq (Submodule R M)] (N : { N // N.FG }) : ↑(Module.fgSystem.equiv R M) ∘ₗ Module.DirectLimit.of R { N // N.FG } (fun i => ↥↑i) (Module.fgSystem R M) N = (↑N).subtype
• Submodule.FG.exists_rTensor_fg_inclusion_eq 📋 Mathlib.Algebra.Colimit.TensorProduct
  {R : Type u_1} {M : Type u_2} {P : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P] {N : Submodule R P} (hN : N.FG) {x y : TensorProduct R (↥N) M} (eq : (LinearMap.rTensor M N.subtype) x = (LinearMap.rTensor M N.subtype) y) : ∃ N', N'.FG ∧ ∃ (h : N ≤ N'), (LinearMap.rTensor M (Submodule.inclusion h)) x = (LinearMap.rTensor M (Submodule.inclusion h)) y
• lTensor_mkQ 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (Q : Type u_5) [AddCommGroup Q] [Module R Q] (N : Submodule R M) : (LinearMap.lTensor Q N.mkQ).ker = (LinearMap.lTensor Q N.subtype).range
• rTensor_mkQ 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (Q : Type u_5) [AddCommGroup Q] [Module R Q] (N : Submodule R M) : (LinearMap.rTensor Q N.mkQ).ker = (LinearMap.rTensor Q N.subtype).range
• LinearMap.lTensor_range 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} (Q : Type u_5) [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] {g : N →ₗ[R] P} : (LinearMap.lTensor Q g).range = (LinearMap.lTensor Q g.range.subtype).range
• LinearMap.rTensor_range 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} (Q : Type u_5) [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] {g : N →ₗ[R] P} : (LinearMap.rTensor Q g).range = (LinearMap.rTensor Q g.range.subtype).range
• Ideal.map_includeRight_eq 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} [CommSemiring R] {A : Type u_2} {B : Type u_3} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (I : Ideal B) : Submodule.restrictScalars R (Ideal.map Algebra.TensorProduct.includeRight I) = (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype).range
• Ideal.map_includeLeft_eq 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} [CommSemiring R] {A : Type u_2} {B : Type u_3} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (I : Ideal A) : Submodule.restrictScalars R (Ideal.map Algebra.TensorProduct.includeLeft I) = (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype).range
• TensorProduct.AlgebraTensorModule.range_lTensor_idealMap 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
  {R : Type u_1} [CommSemiring R] (A : Type u_2) {B : Type u_3} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (S : Type u_4) [CommSemiring S] [Algebra R S] [Algebra S A] [IsScalarTower R S A] (I : Ideal B) : ((TensorProduct.AlgebraTensorModule.lTensor S A) (↑R (Submodule.subtype I))).range = Submodule.restrictScalars S (Ideal.map Algebra.TensorProduct.includeRight I)
• TensorProduct.exists_finite_submodule_left_of_setFinite 📋 Mathlib.LinearAlgebra.TensorProduct.Finiteness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (s : Set (TensorProduct R M N)) (hs : s.Finite) : ∃ M', Module.Finite R ↥M' ∧ s ⊆ ↑(LinearMap.rTensor N M'.subtype).range
• TensorProduct.exists_finite_submodule_right_of_setFinite 📋 Mathlib.LinearAlgebra.TensorProduct.Finiteness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (s : Set (TensorProduct R M N)) (hs : s.Finite) : ∃ N', Module.Finite R ↥N' ∧ s ⊆ ↑(LinearMap.lTensor M N'.subtype).range
• Module.Flat.iff_lTensor_injectiveₛ 📋 Mathlib.RingTheory.Flat.Basic
  {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] : Module.Flat R M ↔ ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Function.Injective ⇑(LinearMap.lTensor M N.subtype)
• Module.Flat.iff_rTensor_injectiveₛ 📋 Mathlib.RingTheory.Flat.Basic
  {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] : Module.Flat R M ↔ ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Function.Injective ⇑(LinearMap.rTensor M N.subtype)
• Module.Flat.mk 📋 Mathlib.RingTheory.Flat.Basic
  {R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (out : ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] [Module.Finite R P] (N : Submodule R P), N.FG → Function.Injective ⇑(LinearMap.rTensor M N.subtype)) : Module.Flat R M
• Module.Flat.out 📋 Mathlib.RingTheory.Flat.Basic
  {R : Type u} {M : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid M} {inst✝² : Module R M} [self : Module.Flat R M] ⦃P : Type u⦄ [AddCommMonoid P] [Module R P] [Module.Finite R P] (N : Submodule R P) : N.FG → Function.Injective ⇑(LinearMap.rTensor M N.subtype)
• Module.flat_iff 📋 Mathlib.RingTheory.Flat.Basic
  (R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M] : Module.Flat R M ↔ ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] [Module.Finite R P] (N : Submodule R P), N.FG → Function.Injective ⇑(LinearMap.rTensor M N.subtype)
• LinearMap.rTensor_injective_iff_subtype 📋 Mathlib.RingTheory.Flat.Basic
  {R : Type u} {M : Type v} {N : Type u_1} {P : Type u_2} {Q : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [AddCommMonoid Q] [Module R Q] {f : N →ₗ[R] P} (hf : Function.Injective ⇑f) (e : P ≃ₗ[R] Q) : Function.Injective ⇑(LinearMap.rTensor M f) ↔ Function.Injective ⇑(LinearMap.rTensor M (↑e ∘ₗ f).range.subtype)
• LinearMap.subtype_compMultilinearMap_codRestrict 📋 Mathlib.LinearAlgebra.Multilinear.Basic
  {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p) : p.subtype.compMultilinearMap (f.codRestrict p h) = f
• LinearMap.subtype_compAlternatingMap_codRestrict 📋 Mathlib.LinearAlgebra.Alternating.Basic
  {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p) : p.subtype.compAlternatingMap (f.codRestrict p h) = f
• ModuleCat.mono_as_hom'_subtype 📋 Mathlib.Algebra.Category.ModuleCat.EpiMono
  {R : Type u} [Ring R] {X : ModuleCat R} (U : Submodule R ↑X) : CategoryTheory.Mono (ModuleCat.ofHom U.subtype)
• LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective 📋 Mathlib.LinearAlgebra.Dual.Lemmas
  {R : Type u_1} {M : Type u_2} {M' : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Surjective ⇑f.range.subtype.dualMap) : f.dualMap.range = f.ker.dualAnnihilator
• ModuleCat.kernelIsoKer_hom_ker_subtype 📋 Mathlib.Algebra.Category.ModuleCat.Kernels
  {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.kernelIsoKer f).hom (ModuleCat.ofHom (ModuleCat.Hom.hom f).ker.subtype) = CategoryTheory.Limits.kernel.ι f
• ModuleCat.kernelIsoKer_inv_kernel_ι 📋 Mathlib.Algebra.Category.ModuleCat.Kernels
  {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.kernelIsoKer f).inv (CategoryTheory.Limits.kernel.ι f) = ModuleCat.ofHom (ModuleCat.Hom.hom f).ker.subtype
• Module.Baer.iff_surjective 📋 Mathlib.Algebra.Module.Injective
  {M : Type u_1} [AddCommGroup M] {R : Type u} [CommRing R] [Module R M] : Module.Baer R M ↔ ∀ (I : Ideal R), Function.Surjective ⇑(LinearMap.lcomp R M (Submodule.subtype I))
• CategoryTheory.ShortComplex.moduleCatLeftHomologyData_i_hom 📋 Mathlib.Algebra.Homology.ShortComplex.ModuleCat
  {R : Type u} [Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)) : ModuleCat.Hom.hom S.moduleCatLeftHomologyData.i = (ModuleCat.Hom.hom S.g).ker.subtype
• TensorProduct.quotientTensorEquiv 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} (N : Type u_3) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (m : Submodule R M) : TensorProduct R (M ⧸ m) N ≃ₗ[R] TensorProduct R M N ⧸ (TensorProduct.map m.subtype LinearMap.id).range
• TensorProduct.tensorQuotientEquiv 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} (M : Type u_2) {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (n : Submodule R N) : TensorProduct R M (N ⧸ n) ≃ₗ[R] TensorProduct R M N ⧸ (TensorProduct.map LinearMap.id n.subtype).range
• TensorProduct.quotientTensorQuotientEquiv 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (m : Submodule R M) (n : Submodule R N) : TensorProduct R (M ⧸ m) (N ⧸ n) ≃ₗ[R] TensorProduct R M N ⧸ (TensorProduct.map m.subtype LinearMap.id).range ⊔ (TensorProduct.map LinearMap.id n.subtype).range
• TensorProduct.quotientTensorEquiv_apply_tmul_mk 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (m : Submodule R M) (x : M) (y : N) : (TensorProduct.quotientTensorEquiv N m) (Submodule.Quotient.mk x ⊗ₜ[R] y) = Submodule.Quotient.mk (x ⊗ₜ[R] y)
• TensorProduct.tensorQuotientEquiv_apply_mk_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (n : Submodule R N) (x : M) (y : N) : (TensorProduct.tensorQuotientEquiv M n) (x ⊗ₜ[R] Submodule.Quotient.mk y) = Submodule.Quotient.mk (x ⊗ₜ[R] y)
• TensorProduct.quotientTensorEquiv_symm_apply_mk_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (m : Submodule R M) (x : M) (y : N) : (TensorProduct.quotientTensorEquiv N m).symm (Submodule.Quotient.mk (x ⊗ₜ[R] y)) = Submodule.Quotient.mk x ⊗ₜ[R] y
• TensorProduct.tensorQuotientEquiv_symm_apply_tmul_mk 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (n : Submodule R N) (x : M) (y : N) : (TensorProduct.tensorQuotientEquiv M n).symm (Submodule.Quotient.mk (x ⊗ₜ[R] y)) = x ⊗ₜ[R] Submodule.Quotient.mk y
• TensorProduct.quotientTensorQuotientEquiv_apply_tmul_mk_tmul_mk 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (m : Submodule R M) (n : Submodule R N) (x : M) (y : N) : (TensorProduct.quotientTensorQuotientEquiv m n) (Submodule.Quotient.mk x ⊗ₜ[R] Submodule.Quotient.mk y) = Submodule.Quotient.mk (x ⊗ₜ[R] y)
• TensorProduct.AlgebraTensorModule.tensorQuotientEquiv 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] (M : Type u_4) [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M] {N : Type u_5} [AddCommGroup N] [Module R N] [Module B N] [IsScalarTower R B N] (n : Submodule B N) : TensorProduct R M (N ⧸ n) ≃ₗ[A] TensorProduct R M N ⧸ ((TensorProduct.AlgebraTensorModule.lTensor A M) (↑R n.subtype)).range
• TensorProduct.quotientTensorQuotientEquiv_symm_apply_mk_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (m : Submodule R M) (n : Submodule R N) (x : M) (y : N) : (TensorProduct.quotientTensorQuotientEquiv m n).symm (Submodule.Quotient.mk (x ⊗ₜ[R] y)) = Submodule.Quotient.mk x ⊗ₜ[R] Submodule.Quotient.mk y
• TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_apply_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] (M : Type u_4) [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M] {N : Type u_5} [AddCommGroup N] [Module R N] [Module B N] [IsScalarTower R B N] (n : Submodule B N) (x : M) (y : N) : (TensorProduct.AlgebraTensorModule.tensorQuotientEquiv A B M n) (x ⊗ₜ[R] Submodule.Quotient.mk y) = Submodule.Quotient.mk (x ⊗ₜ[R] y)
• TensorProduct.AlgebraTensorModule.tensorQuotientEquiv_symm_apply_mk_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
  {R : Type u_1} (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [Algebra R A] [CommRing B] [Algebra R B] (M : Type u_4) [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M] {N : Type u_5} [AddCommGroup N] [Module R N] [Module B N] [IsScalarTower R B N] (n : Submodule B N) (x : M) (y : N) : (TensorProduct.AlgebraTensorModule.tensorQuotientEquiv A B M n).symm (Submodule.Quotient.mk (x ⊗ₜ[R] y)) = x ⊗ₜ[R] Submodule.Quotient.mk y
• Ideal.subtype_rTensor_range 📋 Mathlib.RingTheory.TensorProduct.Quotient
  {R : Type u_1} [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (I : Ideal R) : (↑(TensorProduct.lid R M) ∘ₗ LinearMap.rTensor M (Submodule.subtype I)).range = I • ⊤
• covBy_iff_quot_is_simple 📋 Mathlib.RingTheory.SimpleModule.Basic
  {R : Type u_2} [Ring R] {M : Type u_4} [AddCommGroup M] [Module R M] {A B : Submodule R M} (hAB : A ≤ B) : A ⋖ B ↔ IsSimpleModule R (↥B ⧸ Submodule.comap B.subtype A)
• JordanHolderLattice.Iso.linearEquiv 📋 Mathlib.RingTheory.SimpleModule.Basic
  {R : Type u_2} [Ring R] {M : Type u_4} [AddCommGroup M] [Module R M] {X Y : Submodule R M × Submodule R M} (h : JordanHolderLattice.Iso X Y) : (↥X.2 ⧸ Submodule.comap X.2.subtype X.1) ≃ₗ[R] ↥Y.2 ⧸ Submodule.comap Y.2.subtype Y.1
• Ideal.map_toCotangent_ker 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] (I : Ideal R) : Submodule.map (Submodule.subtype I) I.toCotangent.ker = I ^ 2
• Ideal.mapCotangent_ker_of_surjective 📋 Mathlib.RingTheory.Ideal.Cotangent
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (surj : Function.Surjective ⇑(algebraMap A B)) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J) : (J.mapCotangent I (Algebra.ofId A B) ⋯).ker = Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (RingHom.ker (algebraMap A B) ⊓ J))
• Ideal.to_quotient_square_comp_toCotangent 📋 Mathlib.RingTheory.Ideal.Cotangent
  {R : Type u} [CommRing R] (I : Ideal R) : I.cotangentToQuotientSquare ∘ₗ I.toCotangent = Submodule.mkQ (I ^ 2) ∘ₗ Submodule.subtype I
• DirectSum.decompose_lhom_ext 📋 Mathlib.Algebra.DirectSum.Decomposition
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) [DirectSum.Decomposition ℳ] {N : Type u_5} [AddCommMonoid N] [Module R N] ⦃f g : M →ₗ[R] N⦄ (h : ∀ (i : ι), f ∘ₗ (ℳ i).subtype = g ∘ₗ (ℳ i).subtype) : f = g
• DirectSum.decomposeLinearEquiv_symm_comp_lof 📋 Mathlib.Algebra.DirectSum.Decomposition
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} [DecidableEq ι] [Semiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) [DirectSum.Decomposition ℳ] (i : ι) : ↑(DirectSum.decomposeLinearEquiv ℳ).symm ∘ₗ DirectSum.lof R ι (fun x => ↥(ℳ x)) i = (ℳ i).subtype
• exteriorPower.subtype_comp_map_eq 📋 Mathlib.LinearAlgebra.ExteriorPower.Basic
  {R : Type u} [CommRing R] {n : ℕ} {M : Type u_1} {N : Type u_2} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) : (⋀[R]^n N).subtype ∘ₗ exteriorPower.map n f = (ExteriorAlgebra.map f).toLinearMap ∘ₗ (⋀[R]^n M).subtype
• exteriorPower.ιMulti_family_span 📋 Mathlib.LinearAlgebra.ExteriorPower.Basic
  (R : Type u) [CommRing R] {n : ℕ} {M : Type u_1} [AddCommGroup M] [Module R M] {I : Type u_4} [LinearOrder I] (v : I → M) : (exteriorPower.map n (Submodule.span R (Set.range v)).subtype).range = Submodule.span R (Set.range (exteriorPower.ιMulti_family R n v))
• ModuleCat.imageIsoRange_hom_subtype 📋 Mathlib.Algebra.Category.ModuleCat.Images
  {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.imageIsoRange f).hom (ModuleCat.ofHom (ModuleCat.Hom.hom f).range.subtype) = CategoryTheory.Limits.image.ι f
• ModuleCat.imageIsoRange_inv_image_ι 📋 Mathlib.Algebra.Category.ModuleCat.Images
  {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) : CategoryTheory.CategoryStruct.comp (ModuleCat.imageIsoRange f).inv (CategoryTheory.Limits.image.ι f) = ModuleCat.ofHom (ModuleCat.Hom.hom f).range.subtype
• ModuleCat.imageIsoRange_hom_subtype_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Images
  {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) {Z : ModuleCat R} (h : ModuleCat.of R ↑H ⟶ Z) : CategoryTheory.CategoryStruct.comp (ModuleCat.imageIsoRange f).hom (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (ModuleCat.Hom.hom f).range.subtype) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.ι f) h
• ModuleCat.imageIsoRange_inv_image_ι_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Images
  {R : Type u} [Ring R] {G H : ModuleCat R} (f : G ⟶ H) {Z : ModuleCat R} (h : H ⟶ Z) : CategoryTheory.CategoryStruct.comp (ModuleCat.imageIsoRange f).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.image.ι f) h) = CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom (ModuleCat.Hom.hom f).range.subtype) h
• PresheafOfModules.Submodule.ι_app_hom_apply 📋 Mathlib.Algebra.Category.ModuleCat.Presheaf.Submodule
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {M : PresheafOfModules R} (N : M.Submodule) (X : Cᵒᵖ) (a : ↑(N.toPresheafOfModules.presheaf.obj X)) : (ModuleCat.Hom.hom (N.ι.app X)) a = (CategoryTheory.ConcreteCategory.hom (AddCommGrpCat.ofHom (N.obj X).subtype.toAddMonoidHom)) a
• Submodule.toLinearMap_subtypeL 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) : ↑p.subtypeL = p.subtype
• Submodule.isEmbedding_subtype 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) : Topology.IsEmbedding ⇑p.subtype
• Submodule.isClosedEmbedding_subtype 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) (hp : IsClosed ↑p) : Topology.IsClosedEmbedding ⇑p.subtype
• Submodule.coe_subtypeL 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) : ⇑p.subtypeL = ⇑p.subtype
• Submodule.coe_subtypeL' 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Restrict
  {R : Type u_1} [Semiring R] {M : Type u_2} [TopologicalSpace M] [AddCommMonoid M] [Module R M] (p : Submodule R M) : ⇑p.subtypeL = ⇑p.subtype
• Module.Flat.ker_lTensor_eq 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (M : Type u_3) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} {P : Type u_5} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (f : N →ₗ[R] P) [Module.Flat R M] : ((TensorProduct.AlgebraTensorModule.lTensor S M) f).ker = ((TensorProduct.AlgebraTensorModule.lTensor S M) f.ker.subtype).range
• Module.Flat.eqLocus_lTensor_eq 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (M : Type u_3) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} {P : Type u_5} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (f g : N →ₗ[R] P) [Module.Flat R M] : ((TensorProduct.AlgebraTensorModule.lTensor S M) f).eqLocus ((TensorProduct.AlgebraTensorModule.lTensor S M) g) = ((TensorProduct.AlgebraTensorModule.lTensor S M) (f.eqLocus g).subtype).range
• LinearMap.tensorKer_coe 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (M : Type u_3) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} {P : Type u_5} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (f : N →ₗ[R] P) (x : TensorProduct R M ↥f.ker) : ↑((LinearMap.tensorKer S M f) x) = (LinearMap.lTensor M f.ker.subtype) x
• LinearMap.tensorEqLocus_coe 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (M : Type u_3) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} {P : Type u_5} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (f g : N →ₗ[R] P) (x : TensorProduct R M ↥(f.eqLocus g)) : ↑((LinearMap.tensorEqLocus S M f g) x) = (LinearMap.lTensor M (f.eqLocus g).subtype) x
• LinearMap.lTensor_ker_subtype_tensorKerEquiv_symm 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (M : Type u_3) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} {P : Type u_5} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (f : N →ₗ[R] P) [Module.Flat R M] (x : ↥((TensorProduct.AlgebraTensorModule.lTensor S M) f).ker) : (LinearMap.lTensor M f.ker.subtype) ((LinearMap.tensorKerEquiv S M f).symm x) = ↑x
• LinearMap.lTensor_eqLocus_subtype_tensoreqLocusEquiv_symm 📋 Mathlib.RingTheory.Flat.Equalizer
  {R : Type u_1} (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (M : Type u_3) [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} {P : Type u_5} [AddCommGroup N] [AddCommGroup P] [Module R N] [Module R P] (f g : N →ₗ[R] P) [Module.Flat R M] (x : ↥(((TensorProduct.AlgebraTensorModule.lTensor S M) f).eqLocus ((TensorProduct.AlgebraTensorModule.lTensor S M) g))) : (LinearMap.lTensor M (f.eqLocus g).subtype) ((LinearMap.tensorEqLocusEquiv S M f g).symm x) = ↑x
• Module.End.eigenspace_restrict_le_eigenspace 📋 Mathlib.LinearAlgebra.Eigenspace.Basic
  {R : Type v} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : Module.End R M) {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p) (μ : R) : Submodule.map p.subtype (Module.End.eigenspace (LinearMap.restrict f hfp) μ) ≤ f.eigenspace μ
• Submodule.inf_genEigenspace 📋 Mathlib.LinearAlgebra.Eigenspace.Basic
  {R : Type v} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : Module.End R M) (p : Submodule R M) {k : ℕ∞} {μ : R} (hfp : ∀ x ∈ p, f x ∈ p) : p ⊓ (f.genEigenspace μ) k = Submodule.map p.subtype ((Module.End.genEigenspace (LinearMap.restrict f hfp) μ) k)
• Module.End.genEigenspace_restrict 📋 Mathlib.LinearAlgebra.Eigenspace.Basic
  {R : Type v} {M : Type w} [CommRing R] [AddCommGroup M] [Module R M] (f : Module.End R M) (p : Submodule R M) (k : ℕ∞) (μ : R) (hfp : ∀ x ∈ p, f x ∈ p) : (Module.End.genEigenspace (LinearMap.restrict f hfp) μ) k = Submodule.comap p.subtype ((f.genEigenspace μ) k)
• LieSubmodule.incl_coe 📋 Mathlib.Algebra.Lie.Submodule
  {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (N : LieSubmodule R L M) : ↑N.incl = (↑N).subtype
• LieSubmodule.toEnd_comp_subtype_mem 📋 Mathlib.Algebra.Lie.OfAssociative
  {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (x : L) (m : M) (hm : m ∈ ↑N) : ((LieModule.toEnd R L M) x ∘ₗ (↑N).subtype) ⟨m, hm⟩ ∈ ↑N
• LieSubmodule.toEnd_restrict_eq_toEnd 📋 Mathlib.Algebra.Lie.OfAssociative
  {R : Type u} {L : Type v} {M : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] (N : LieSubmodule R L M) (x : L) (h : ∀ (m : M) (hm : m ∈ ↑N), ((LieModule.toEnd R L M) x ∘ₗ (↑N).subtype) ⟨m, hm⟩ ∈ ↑N := ⋯) : LinearMap.restrict ((LieModule.toEnd R L M) x) h = (LieModule.toEnd R L ↥N) x
• LieIdeal.incl_coe 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) : ↑I.incl = (LieIdeal.toLieSubalgebra R L I).subtype
• Submodule.inf_iInf_maxGenEigenspace_of_forall_mapsTo 📋 Mathlib.LinearAlgebra.Eigenspace.Pi
  {ι : Type u_1} {R : Type u_2} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] (f : ι → Module.End R M) {μ : ι → R} (p : Submodule R M) (hfp : ∀ (i : ι), Set.MapsTo ⇑(f i) ↑p ↑p) : p ⊓ ⨅ i, (f i).maxGenEigenspace (μ i) = Submodule.map p.subtype (⨅ i, Module.End.maxGenEigenspace (LinearMap.restrict (f i) ⋯) (μ i))
• Module.End.iInf_maxGenEigenspace_restrict_map_subtype_eq 📋 Mathlib.LinearAlgebra.Eigenspace.Pi
  {ι : Type u_1} {R : Type u_2} {M : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] (f : ι → Module.End R M) {μ : ι → R} (i : ι) (h : ∀ (j : ι), Set.MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))) : Submodule.map ((f i).maxGenEigenspace (μ i)).subtype (⨅ j, (fun j => Module.End.maxGenEigenspace (LinearMap.restrict (f j) ⋯) (μ j)) j) = ⨅ j, (f j).maxGenEigenspace (μ j)
• LinearMap.BilinForm.inf_orthogonal_self_le_ker_restrict 📋 Mathlib.LinearAlgebra.BilinearForm.Orthogonal
  {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} {W : Submodule R M} (b₁ : B.IsRefl) : W ⊓ B.orthogonal W ≤ Submodule.map W.subtype (LinearMap.ker (B.restrict W))
• LinearMap.BilinForm.toLin_restrict_ker_eq_inf_orthogonal 📋 Mathlib.LinearAlgebra.BilinearForm.Orthogonal
  {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] (B : LinearMap.BilinForm K V) (W : Subspace K V) (b : B.IsRefl) : Submodule.map (Submodule.subtype W) (LinearMap.domRestrict B W).ker = W ⊓ B.orthogonal ⊤
• LinearMap.BilinForm.ker_restrict_eq_of_codisjoint 📋 Mathlib.LinearAlgebra.BilinearForm.Orthogonal
  {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} (hpq : Codisjoint p q) {B : LinearMap.BilinForm R M} (hB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0) : LinearMap.ker (B.restrict p) = Submodule.comap p.subtype (LinearMap.ker B)
• LinearMap.BilinForm.toLin_restrict_ker_eq_inf_ker 📋 Mathlib.LinearAlgebra.BilinearForm.Orthogonal
  {V : Type u_5} {K : Type u_6} [Field K] [AddCommGroup V] [Module K V] (B : LinearMap.BilinForm K V) (W : Subspace K V) : Submodule.map (Submodule.subtype W) (LinearMap.domRestrict B W).ker = W ⊓ LinearMap.ker B
• LieDerivation.IsKilling.ad_mem_ker_killingForm_ad_range_of_mem_orthogonal 📋 Mathlib.Algebra.Lie.Derivation.Killing
  {R : Type u_1} {L : Type u_2} [Field R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] {D : LieDerivation R L L} (hD : D ∈ (killingForm R (LieDerivation R L L)).orthogonal (LieDerivation.ad R L).range.toSubmodule) (x : L) : (LieDerivation.ad R L) (D x) ∈ Submodule.map (LieDerivation.ad R L).range.subtype (LinearMap.ker (killingForm R ↥(LieDerivation.ad R L).range))
• Module.injOn_dualMap_subtype_span_range_range 📋 Mathlib.LinearAlgebra.Reflection
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {ι : Type u_3} [IsAddTorsionFree M] {r : ι ↪ M} {c : ι → Module.Dual R M} (hfin : (Set.range ⇑r).Finite) (h_two : ∀ (i : ι), (c i) (r i) = 2) (h_mapsTo : ∀ (i : ι), Set.MapsTo (⇑(Module.preReflection (r i) (c i))) (Set.range ⇑r) (Set.range ⇑r)) : Set.InjOn (⇑(Submodule.span R (Set.range ⇑r)).subtype.dualMap) (Set.range c)
• Module.Dual.eq_of_preReflection_mapsTo' 📋 Mathlib.LinearAlgebra.Reflection
  {R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [CharZero R] [IsDomain R] [Module.IsTorsionFree R M] {x : M} {Φ : Set M} (hΦ₁ : Φ.Finite) (hx : x ∈ Submodule.span R Φ) {f g : Module.Dual R M} (hf₁ : f x = 2) (hf₂ : Set.MapsTo (⇑(Module.preReflection x f)) Φ Φ) (hg₁ : g x = 2) (hg₂ : Set.MapsTo (⇑(Module.preReflection x g)) Φ Φ) : (Submodule.span R Φ).subtype.dualMap f = (Submodule.span R Φ).subtype.dualMap g
• Module.Flat.iff_lTensor_injective' 📋 Mathlib.RingTheory.Flat.Tensor
  {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(LinearMap.lTensor M (Submodule.subtype I))
• Module.Flat.iff_rTensor_injective' 📋 Mathlib.RingTheory.Flat.Tensor
  {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype I))
• Module.Flat.iff_lTensor_injective 📋 Mathlib.RingTheory.Flat.Tensor
  {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(LinearMap.lTensor M (Submodule.subtype I))
• Module.Flat.iff_rTensor_injective 📋 Mathlib.RingTheory.Flat.Tensor
  {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype I))
• Module.Flat.iff_lift_lsmul_comp_subtype_injective 📋 Mathlib.RingTheory.Flat.Tensor
  {R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(TensorProduct.lift (LinearMap.lsmul R M ∘ₗ Submodule.subtype I))
• Ideal.subtype_isoBaseOfIsPrincipal_eq_mul 📋 Mathlib.RingTheory.Ideal.IsPrincipal
  {R : Type u_1} [CommRing R] [IsDomain R] {I : Ideal R} [hprinc : Submodule.IsPrincipal I] (h : I ≠ ⊥) : Submodule.subtype I ∘ₗ ↑(Ideal.isoBaseOfIsPrincipal h) = (LinearMap.mul R R) (Submodule.IsPrincipal.generator I)
• RootPairing.injOn_dualMap_subtype_span_root_coroot 📋 Mathlib.LinearAlgebra.RootSystem.Basic
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) [Finite ι] [IsAddTorsionFree M] : Set.InjOn (⇑((Submodule.span R (Set.range ⇑P.root)).subtype.dualMap ∘ₗ P.flip)) (Set.range ⇑P.coroot)
• DirectSum.decomposeTensor_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Decomposition
  {ι : Type u_1} {R : Type u_2} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) (N : Type u_5) [AddCommMonoid N] [Module R N] {i : ι} : DirectSum.decomposeTensor ℳ N i = (LinearMap.rTensor N (ℳ i).subtype).range

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59