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Found 91 declarations mentioning Submodule.toAddSubmonoid.

Submodule.toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (self : Submodule R M) : AddSubmonoid M
Submodule.toAddSubmonoid_injective 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : Function.Injective Submodule.toAddSubmonoid
Submodule.toAddSubmonoid_inj 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : p.toAddSubmonoid = q.toAddSubmonoid ↔ p = q
Submodule.coe_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) : ↑p.toAddSubmonoid = ↑p
Submodule.carrier_eq_coe 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (s : Submodule R M) : s.carrier = ↑s
Submodule.mem_carrier 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} : x ∈ p.carrier ↔ x ∈ ↑p
Submodule.mem_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p : Submodule R M) (x : M) : x ∈ p.toAddSubmonoid ↔ x ∈ p
Submodule.smul_mem' 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (self : Submodule R M) (c : R) {x : M} : x ∈ self.carrier → c • x ∈ self.carrier
Submodule.eta 📋 Mathlib.Algebra.Module.Submodule.Defs
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M} (h : ∀ (c : R) {x : M}, x ∈ p.carrier → c • x ∈ p.carrier) : { toAddSubmonoid := p.toAddSubmonoid, smul_mem' := h } = p
Submodule.toAddSubmonoid_mono 📋 Mathlib.Algebra.Module.Submodule.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : Monotone Submodule.toAddSubmonoid
Submodule.toAddSubmonoid_strictMono 📋 Mathlib.Algebra.Module.Submodule.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] : StrictMono Submodule.toAddSubmonoid
Submodule.toAddSubgroup_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Basic
{R : Type u} {M : Type v} [Ring R] [AddCommGroup M] {module_M : Module R M} (p : Submodule R M) : p.toAddSubgroup.toAddSubmonoid = p.toAddSubmonoid
Submodule.toAddSubmonoid_le 📋 Mathlib.Algebra.Module.Submodule.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {p q : Submodule R M} : p.toAddSubmonoid ≤ q.toAddSubmonoid ↔ p ≤ q
Submodule.bot_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Lattice
{R : Type u_1} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] : ⊥.toAddSubmonoid = ⊥
Submodule.top_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Lattice
{R : Type u_1} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] : ⊤.toAddSubmonoid = ⊤
Submodule.toAddSubmonoid_sSup 📋 Mathlib.Algebra.Module.Submodule.Lattice
{R : Type u_1} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set (Submodule R M)) : (sSup s).toAddSubmonoid = sSup (Submodule.toAddSubmonoid '' s)
AddSubmonoid.toNatSubmodule_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Lattice
{M : Type u_3} [AddCommMonoid M] (S : AddSubmonoid M) : (AddSubmonoid.toNatSubmodule S).toAddSubmonoid = S
Submodule.toAddSubmonoid_toNatSubmodule 📋 Mathlib.Algebra.Module.Submodule.Lattice
{M : Type u_3} [AddCommMonoid M] (S : Submodule ℕ M) : AddSubmonoid.toNatSubmodule S.toAddSubmonoid = S
AddSubmonoid.toNatSubmodule_symm 📋 Mathlib.Algebra.Module.Submodule.Lattice
{M : Type u_3} [AddCommMonoid M] : ⇑AddSubmonoid.toNatSubmodule.symm = Submodule.toAddSubmonoid
Submodule.map_toAddSubmonoid' 📋 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] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) : (Submodule.map f p).toAddSubmonoid = AddSubmonoid.map f p.toAddSubmonoid
Submodule.map_toAddSubmonoid 📋 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] {σ₁₂ : R →+* R₂} [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) : (Submodule.map f p).toAddSubmonoid = AddSubmonoid.map (↑f) p.toAddSubmonoid
Submodule.toAddSubmonoid_restrictScalars 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
(S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (V : Submodule R M) : (Submodule.restrictScalars S V).toAddSubmonoid = V.toAddSubmonoid
LinearMap.ker_toAddSubmonoid 📋 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.ker.toAddSubmonoid = AddMonoidHom.mker f
LinearMap.range_toAddSubmonoid 📋 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.toAddSubmonoid = AddMonoidHom.mrange f
Submodule.neg_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Pointwise
{R : Type u_2} {M : Type u_3} [Semiring R] [AddCommGroup M] [Module R M] (S : Submodule R M) : (-S).toAddSubmonoid = -S.toAddSubmonoid
Submodule.pointwise_smul_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.Pointwise
{α : Type u_1} {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid α] [DistribMulAction α M] [SMulCommClass α R M] (a : α) (S : Submodule R M) : (a • S).toAddSubmonoid = a • S.toAddSubmonoid
Submodule.span_nat_eq_addSubmonoidClosure 📋 Mathlib.LinearAlgebra.Span.Defs
{M : Type u_2} [AddCommMonoid M] (s : Set M) : (Submodule.span ℕ s).toAddSubmonoid = AddSubmonoid.closure s
Submodule.span_nat_eq 📋 Mathlib.LinearAlgebra.Span.Defs
{M : Type u_2} [AddCommMonoid M] (s : AddSubmonoid M) : (Submodule.span ℕ ↑s).toAddSubmonoid = s
Submodule.closure_le_toAddSubmonoid_span 📋 Mathlib.LinearAlgebra.Span.Defs
{R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M} : AddSubmonoid.closure s ≤ (Submodule.span R s).toAddSubmonoid
Submodule.span_eq_closure 📋 Mathlib.LinearAlgebra.Span.Defs
{R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M} : (Submodule.span R s).toAddSubmonoid = AddSubmonoid.closure (Set.univ • s)
Submodule.sup_toAddSubmonoid 📋 Mathlib.LinearAlgebra.Span.Defs
{R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (p p' : Submodule R M) : (p ⊔ p').toAddSubmonoid = p.toAddSubmonoid ⊔ p'.toAddSubmonoid
LinearMap.eqLocus_toAddSubmonoid 📋 Mathlib.Algebra.Module.Submodule.EqLocus
{R : Type u_1} {R₂ : Type u_2} {M : Type u_3} {M₂ : Type u_4} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {τ₁₂ : R →+* R₂} (f g : M →ₛₗ[τ₁₂] M₂) : (f.eqLocus g).toAddSubmonoid = (↑f).eqLocusM ↑g
Submodule.iSup_toAddSubmonoid 📋 Mathlib.LinearAlgebra.Span.Basic
{R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Sort u_8} (p : ι → Submodule R M) : (⨆ i, p i).toAddSubmonoid = ⨆ i, (p i).toAddSubmonoid
Submodule.orderIsoOfAlgebraMapSurjective_symm_apply 📋 Mathlib.Algebra.Algebra.Tower
{R : Type u_1} {S : Type u_2} {M : Type u_3} [CommRing R] [Ring S] [AddCommGroup M] [Algebra R S] [Module R M] [Module S M] [IsScalarTower R S M] (h : Function.Surjective ⇑(algebraMap R S)) (N : Submodule R M) : (RelIso.symm (Submodule.orderIsoOfAlgebraMapSurjective h)) N = { toAddSubmonoid := N.toAddSubmonoid, smul_mem' := ⋯ }
Submodule.fg_iff_addSubmonoid_fg 📋 Mathlib.RingTheory.Finiteness.Defs
{M : Type u_2} [AddCommMonoid M] (P : Submodule ℕ M) : P.FG ↔ P.FG
Submodule.le_one_toAddSubmonoid 📋 Mathlib.Algebra.Algebra.Operations
{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A] : 1 ≤ Submodule.toAddSubmonoid 1
Submodule.pow_toAddSubmonoid 📋 Mathlib.Algebra.Algebra.Operations
{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A] [IsScalarTower R A A] (M : Submodule R A) {n : ℕ} (h : n ≠ 0) : (M ^ n).toAddSubmonoid = M.toAddSubmonoid ^ n
Submodule.mul_toAddSubmonoid 📋 Mathlib.Algebra.Algebra.Operations
{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A] [IsScalarTower R A A] (M N : Submodule R A) : (M * N).toAddSubmonoid = M.toAddSubmonoid * N.toAddSubmonoid
Submodule.le_pow_toAddSubmonoid 📋 Mathlib.Algebra.Algebra.Operations
{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A] [IsScalarTower R A A] (M : Submodule R A) {n : ℕ} : M.toAddSubmonoid ^ n ≤ (M ^ n).toAddSubmonoid
Submodule.smul_toAddSubmonoid 📋 Mathlib.Algebra.Algebra.Operations
{R : Type u} [Semiring R] {A : Type v} [Semiring A] [Module R A] {M : Type u_1} [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] {I : Submodule R A} {N : Submodule R M} : (I • N).toAddSubmonoid = I.toAddSubmonoid • N.toAddSubmonoid
Submodule.coe_toSubalgebra 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (p : Submodule R A) (h_one : 1 ∈ p) (h_mul : ∀ (x y : A), x ∈ p → y ∈ p → x * y ∈ p) : ↑(p.toSubalgebra h_one h_mul) = p.carrier
Submodule.toSubalgebra_toSubsemiring 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u_1} {A : Type u_2} [CommSemiring R] [Semiring A] [Algebra R A] (p : Submodule R A) (h_one : 1 ∈ p) (h_mul : ∀ (x y : A), x ∈ p → y ∈ p → x * y ∈ p) : (p.toSubalgebra h_one h_mul).toSubsemiring = { carrier := p.carrier, mul_mem' := ⋯, one_mem' := h_one, add_mem' := ⋯, zero_mem' := ⋯ }
Subalgebra.coe_pi 📋 Mathlib.Algebra.Algebra.Subalgebra.Pi
{ι : Type u_1} {R : Type u_2} {S : ι → Type u_3} [CommSemiring R] [(i : ι) → Semiring (S i)] [(i : ι) → Algebra R (S i)] (s : Set ι) (t : (i : ι) → Subalgebra R (S i)) : ↑(Subalgebra.pi s t) = (Submodule.pi s fun i => Subalgebra.toSubmodule (t i)).carrier
Subalgebra.pi_toSubsemiring 📋 Mathlib.Algebra.Algebra.Subalgebra.Pi
{ι : Type u_1} {R : Type u_2} {S : ι → Type u_3} [CommSemiring R] [(i : ι) → Semiring (S i)] [(i : ι) → Algebra R (S i)] (s : Set ι) (t : (i : ι) → Subalgebra R (S i)) : (Subalgebra.pi s t).toSubsemiring = { carrier := (Submodule.pi s fun i => Subalgebra.toSubmodule (t i)).carrier, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯ }
Ideal.pointwise_smul_toAddSubmonoid 📋 Mathlib.RingTheory.Ideal.Pointwise
{M : Type u_1} {R : Type u_3} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) (S : Ideal R) (ha : Function.Surjective fun r => a • r) : (a • S).toAddSubmonoid = a • S.toAddSubmonoid
Ideal.Filtration.submodule_closure_single 📋 Mathlib.RingTheory.Filtration
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {I : Ideal R} (F : I.Filtration M) : AddSubmonoid.closure (⋃ i, PolynomialModule.single R i '' ↑(F.N i)) = F.submodule.toAddSubmonoid
LieSubalgebra.mem_carrier 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L} : x ∈ L'.carrier ↔ x ∈ ↑L'
LieSubalgebra.lie_mem' 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (self : LieSubalgebra R L) {x y : L} : x ∈ self.carrier → y ∈ self.carrier → ⁅x, y⁆ ∈ self.carrier
LieSubalgebra.mk 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (toSubmodule : Submodule R L) (lie_mem' : ∀ {x y : L}, x ∈ toSubmodule.carrier → y ∈ toSubmodule.carrier → ⁅x, y⁆ ∈ toSubmodule.carrier) : LieSubalgebra R L
LieSubalgebra.toSubmodule_mk 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : Submodule R L) (h : ∀ {x y : L}, x ∈ p.carrier → y ∈ p.carrier → ⁅x, y⁆ ∈ p.carrier) : { toSubmodule := p, lie_mem' := h }.toSubmodule = p
LieSubalgebra.mem_mk_iff' 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : Submodule R L) (h : ∀ {x y : L}, x ∈ p.carrier → y ∈ p.carrier → ⁅x, y⁆ ∈ p.carrier) {x : L} : x ∈ { toSubmodule := p, lie_mem' := h } ↔ x ∈ p
LieSubalgebra.mk_coe 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : R) {x : L}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier → c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ : ∀ {x y : L}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → y ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → ⁅x, y⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) : ↑{ carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } = S
LieSubalgebra.mem_mk_iff 📋 Mathlib.Algebra.Lie.Subalgebra
{R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : R) {x : L}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier → c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ : ∀ {x y : L}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → y ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → ⁅x, y⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) {x : L} : x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } ↔ x ∈ S
LieSubmodule.mem_carrier 📋 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) {x : M} : x ∈ (↑N).carrier ↔ x ∈ ↑N
LieSubmodule.mk 📋 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] (toSubmodule : Submodule R M) (lie_mem : ∀ {x : L} {m : M}, m ∈ toSubmodule.carrier → ⁅x, m⁆ ∈ toSubmodule.carrier) : LieSubmodule R L M
LieSubmodule.lie_mem 📋 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] (self : LieSubmodule R L M) {x : L} {m : M} : m ∈ (↑self).carrier → ⁅x, m⁆ ∈ (↑self).carrier
LieSubmodule.toSubmodule_mk 📋 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] (p : Submodule R M) (h : ∀ {x : L} {m : M}, m ∈ p.carrier → ⁅x, m⁆ ∈ p.carrier) : ↑{ toSubmodule := p, lie_mem := h } = p
LieSubmodule.mk_eq_bot_iff 📋 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 : Submodule R M} {h : ∀ {x : L} {m : M}, m ∈ N.carrier → ⁅x, m⁆ ∈ N.carrier} : { toSubmodule := N, lie_mem := h } = ⊥ ↔ N = ⊥
LieSubmodule.mk_eq_top_iff 📋 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 : Submodule R M} {h : ∀ {x : L} {m : M}, m ∈ N.carrier → ⁅x, m⁆ ∈ N.carrier} : { toSubmodule := N, lie_mem := h } = ⊤ ↔ N = ⊤
LieSubmodule.mem_mk_iff' 📋 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] (p : Submodule R M) (h : ∀ {x : L} {m : M}, m ∈ p.carrier → ⁅x, m⁆ ∈ p.carrier) {x : M} : x ∈ { toSubmodule := p, lie_mem := h } ↔ x ∈ p
LieSubmodule.coe_toSet_mk 📋 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] (S : Set M) (h₁ : ∀ {a b : M}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : R) {x : M}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier → c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ : ∀ {x : L} {m : M}, m ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → ⁅x, m⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) : ↑{ carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem := h₄ } = S
LieSubmodule.mem_mk_iff 📋 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] (S : Set M) (h₁ : ∀ {a b : M}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : R) {x : M}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier → c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ : ∀ {x : L} {m : M}, m ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → ⁅x, m⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) {x : M} : x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem := h₄ } ↔ x ∈ S
HomogeneousIdeal.toIdeal_irrelevant_le 📋 Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
{ι : Type u_1} {σ : Type u_2} {A : Type u_3} [Semiring A] [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜] {I : Ideal A} : (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ I ↔ ∀ i > 0, AddSubmonoid.ofClass (𝒜 i) ≤ I.toAddSubmonoid
HomogeneousIdeal.irrelevant_eq_closure 📋 Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
{ι : Type u_1} {σ : Type u_2} {A : Type u_3} [Semiring A] [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜] : (HomogeneousIdeal.irrelevant 𝒜).toAddSubmonoid = AddSubmonoid.closure (⋃ i, ⋃ (_ : i > 0), ↑(𝒜 i))
HomogeneousIdeal.irrelevant_le 📋 Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
{ι : Type u_1} {σ : Type u_2} {A : Type u_3} [Semiring A] [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜] {P : HomogeneousIdeal 𝒜} : HomogeneousIdeal.irrelevant 𝒜 ≤ P ↔ ∀ i > 0, AddSubmonoid.ofClass (𝒜 i) ≤ P.toAddSubmonoid
HomogeneousIdeal.toAddSubmonoid_irrelevant_le 📋 Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
{ι : Type u_1} {σ : Type u_2} {A : Type u_3} [Semiring A] [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜] {P : AddSubmonoid A} : (HomogeneousIdeal.irrelevant 𝒜).toAddSubmonoid ≤ P ↔ ∀ i > 0, AddSubmonoid.ofClass (𝒜 i) ≤ P
HomogeneousIdeal.irrelevant_eq_iSup 📋 Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
{ι : Type u_1} {σ : Type u_2} {A : Type u_3} [Semiring A] [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ) [GradedRing 𝒜] : (HomogeneousIdeal.irrelevant 𝒜).toAddSubmonoid = ⨆ i, ⨆ (_ : i > 0), AddSubmonoid.ofClass (𝒜 i)
LieRinehartSubalgebra.mem_carrier 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (L' : LieRinehartSubalgebra A L) {x : L} : x ∈ L'.carrier ↔ x ∈ ↑L'
LieRinehartSubalgebra.mk 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (toSubmodule : Submodule A L) (lie_mem' : ∀ {a b : L}, a ∈ toSubmodule.carrier → b ∈ toSubmodule.carrier → ⁅a, b⁆ ∈ toSubmodule.carrier) : LieRinehartSubalgebra A L
LieRinehartSubalgebra.lie_mem' 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (self : LieRinehartSubalgebra A L) {a b : L} : a ∈ self.carrier → b ∈ self.carrier → ⁅a, b⁆ ∈ self.carrier
LieRinehartSubalgebra.toSubmodule_mk 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (p : Submodule A L) (h : ∀ {a b : L}, a ∈ p.carrier → b ∈ p.carrier → ⁅a, b⁆ ∈ p.carrier) : { toSubmodule := p, lie_mem' := h }.toSubmodule = p
LieRinehartSubalgebra.mem_mk_iff' 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (p : Submodule A L) (h : ∀ {a b : L}, a ∈ p.carrier → b ∈ p.carrier → ⁅a, b⁆ ∈ p.carrier) {x : L} : x ∈ { toSubmodule := p, lie_mem' := h } ↔ x ∈ p
LieRinehartSubalgebra.mk_coe 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : A) {x : L}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier → c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ : ∀ {a b : L}, a ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → b ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → ⁅a, b⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) : ↑{ carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } = S
LieRinehartSubalgebra.mem_mk_iff 📋 Mathlib.Algebra.LieRinehartAlgebra.Subalgebra
{A : Type u_1} {L : Type u_2} [CommRing A] [LieRing L] [Module A L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ : ∀ (c : A) {x : L}, x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier → c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ : ∀ {a b : L}, a ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → b ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier → ⁅a, b⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) {x : L} : x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } ↔ x ∈ S
ClosedSubmodule.mk 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [TopologicalSpace M] [Module R M] (toSubmodule : Submodule R M) (isClosed' : IsClosed toSubmodule.carrier) : ClosedSubmodule R M
ClosedSubmodule.isClosed' 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [TopologicalSpace M] [Module R M] (self : ClosedSubmodule R M) : IsClosed (↑self).carrier
ClosedSubmodule.carrier_eq_coe 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [TopologicalSpace M] [Module R M] (s : ClosedSubmodule R M) : (↑s).carrier = ↑s
ClosedSubmodule.ext 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} {inst✝ : Semiring R} {inst✝¹ : AddCommMonoid M} {inst✝² : TopologicalSpace M} {inst✝³ : Module R M} {x y : ClosedSubmodule R M} (carrier : (↑x).carrier = (↑y).carrier) : x = y
ClosedSubmodule.ext_iff 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} {inst✝ : Semiring R} {inst✝¹ : AddCommMonoid M} {inst✝² : TopologicalSpace M} {inst✝³ : Module R M} {x y : ClosedSubmodule R M} : x = y ↔ (↑x).carrier = (↑y).carrier
ClosedSubmodule.mem_mk 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [TopologicalSpace M] [Module R M] {x : M} {s : Submodule R M} {hs : IsClosed s.carrier} : x ∈ { toSubmodule := s, isClosed' := hs } ↔ x ∈ s
Submodule.closure_eq' 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [TopologicalSpace M] [Module R M] [ContinuousAdd M] [ContinuousConstSMul R M] {s : Submodule R M} (hs : IsClosed s.carrier) : s.closure = { toSubmodule := s, isClosed' := hs }
ClosedSubmodule.coe_iSup 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{ι : Sort u_1} {R : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid N] [TopologicalSpace N] [Module R N] [ContinuousAdd N] [ContinuousConstSMul R N] (f : ι → ClosedSubmodule R N) : ↑(⨆ i, f i) = closure (⨆ i, ↑(f i)).carrier
ClosedSubmodule.coe_sup 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid N] [TopologicalSpace N] [Module R N] [ContinuousAdd N] [ContinuousConstSMul R N] {s t : ClosedSubmodule R N} : ↑(s ⊔ t) = closure (↑s ⊔ ↑t).carrier
ClosedSubmodule.mem_iSup 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{ι : Sort u_1} {R : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid N] [TopologicalSpace N] [Module R N] [ContinuousAdd N] [ContinuousConstSMul R N] {x : N} {f : ι → ClosedSubmodule R N} : x ∈ ⨆ i, f i ↔ x ∈ closure (⨆ i, ↑(f i)).carrier
ClosedSubmodule.mem_sup 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid N] [TopologicalSpace N] [Module R N] [ContinuousAdd N] [ContinuousConstSMul R N] {s t : ClosedSubmodule R N} {x : N} : x ∈ s ⊔ t ↔ x ∈ closure (↑s ⊔ ↑t).carrier
ClosedSubmodule.coe_sSup 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid N] [TopologicalSpace N] [Module R N] [ContinuousAdd N] [ContinuousConstSMul R N] (S : Set (ClosedSubmodule R N)) : ↑(sSup S) = closure (⨆ s ∈ S, ↑s).carrier
ClosedSubmodule.mem_sSup 📋 Mathlib.Topology.Algebra.Module.ClosedSubmodule
{R : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid N] [TopologicalSpace N] [Module R N] [ContinuousAdd N] [ContinuousConstSMul R N] {x : N} {S : Set (ClosedSubmodule R N)} : x ∈ sSup S ↔ x ∈ closure (⨆ s ∈ S, ↑s).carrier
PointedCone.support_eq 📋 Mathlib.Geometry.Convex.Cone.Pointed
{R : Type u_1} {E : Type u_2} [Ring R] [LinearOrder R] [IsOrderedRing R] [AddCommGroup E] [Module R E] (C : PointedCone R E) : C.support = C.lineal.toAddSubgroup
ClosedSubmodule.mem_iff 📋 Mathlib.Analysis.InnerProductSpace.StandardSubspace
{H : Type u_1} [NormedAddCommGroup H] [ipc : InnerProductSpace ℂ H] (S : ClosedSubmodule ℝ H) {x : H} : x ∈ S ↔ x ∈ (↑S).carrier
LinearEquiv.map_eq_of_mem_fixingSubgroup 📋 Mathlib.LinearAlgebra.FixedSubmodule
{R : Type u_1} [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (e : V ≃ₗ[R] V) (W : Submodule R V) (he : e ∈ fixingSubgroup (V ≃ₗ[R] V) W.carrier) : Submodule.map (↑e) W = W
Representation.invariants_eq_inter 📋 Mathlib.RepresentationTheory.Invariants
{k : Type u_1} {G : Type u_2} {V : Type u_3} [CommRing k] [Group G] [AddCommGroup V] [Module k V] (ρ : Representation k G V) : ρ.invariants.carrier = ⋂ g, Function.fixedPoints ⇑(ρ g)

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:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
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
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.
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