Loogle!

Result

Found 53 declarations mentioning LinearMap.range and Submodule.span.

• LinearMap.range_toSpanSingleton 📋 Mathlib.LinearAlgebra.Span.Basic
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (x : M) : (LinearMap.toSpanSingleton R M x).range = R ∙ x
• LinearMap.span_singleton_eq_range 📋 Mathlib.LinearAlgebra.Span.Basic
  (R : Type u_1) (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] (x : M) : R ∙ x = (LinearMap.toSpanSingleton R M x).range
• Fintype.range_linearCombination 📋 Mathlib.LinearAlgebra.Finsupp.LinearCombination
  {α : Type u_1} {M : Type u_2} (R : Type u_3) [Fintype α] [Semiring R] [AddCommMonoid M] [Module R M] (v : α → M) : (Fintype.linearCombination R v).range = Submodule.span R (Set.range v)
• Finsupp.range_linearCombination 📋 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} : (Finsupp.linearCombination R v).range = Submodule.span R (Set.range v)
• Finsupp.span_eq_range_linearCombination 📋 Mathlib.LinearAlgebra.Finsupp.LinearCombination
  {M : Type u_2} (R : Type u_3) [Semiring R] [AddCommMonoid M] [Module R M] (s : Set M) : Submodule.span R s = (Finsupp.linearCombination R Subtype.val).range
• Finsupp.linearCombinationOn_range 📋 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.linearCombinationOn α M R v s).range = ⊤
• Module.Basis.constr_range 📋 Mathlib.LinearAlgebra.Basis.Defs
  {M' : Type u_5} [AddCommMonoid M'] {ι : Type u_7} {R : Type u_8} {M : Type u_9} [Semiring R] [AddCommMonoid M] [Module R M] (b : Module.Basis ι R M) [Module R M'] (S : Type u_10) [Semiring S] [Module S M'] [SMulCommClass R S M'] {f : ι → M'} : ((b.constr S) f).range = Submodule.span R (Set.range f)
• LinearIndependent.repr_range 📋 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) : hv.repr.range = ⊤
• LinearIndependent.linearCombinationEquiv_symm_apply 📋 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) (a✝ : ↥(Submodule.span R (Set.range v))) : hv.linearCombinationEquiv.symm a✝ = ((LinearEquiv.ofInjective (LinearMap.codRestrict (Submodule.span R (Set.range v)) (Finsupp.linearCombination R v) ⋯) ⋯).toEquiv.trans (LinearEquiv.ofTop (LinearMap.codRestrict (Submodule.span R (Set.range v)) (Finsupp.linearCombination R v) ⋯).range ⋯).toEquiv).invFun a✝
• Submodule.span_preimage_eq 📋 Mathlib.LinearAlgebra.Quotient.Basic
  {R : Type u_1} {M : Type u_2} [Ring R] [AddCommGroup M] [Module R M] {R₂ : Type u_3} {M₂ : Type u_4} [Ring R₂] [AddCommGroup M₂] [Module R₂ M₂] {τ₁₂ : R →+* R₂} [RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {s : Set M₂} (h₀ : s.Nonempty) (h₁ : s ⊆ ↑f.range) : Submodule.span R (⇑f ⁻¹' s) = Submodule.comap f (Submodule.span R₂ s)
• LinearMap.range_dualMap_dual_eq_span_singleton 📋 Mathlib.LinearAlgebra.Dual.Defs
  {R : Type u_1} {M₁ : Type u_2} [CommSemiring R] [AddCommMonoid M₁] [Module R M₁] (f : Module.Dual R M₁) : (LinearMap.dualMap f).range = R ∙ f
• Finsupp.range_lmapDomain 📋 Mathlib.LinearAlgebra.Finsupp.Span
  {α : Type u_1} {R : Type u_3} [Semiring R] {β : Type u_5} (u : α → β) : (Finsupp.lmapDomain R R u).range = Submodule.span R (Set.range fun x => fun₀ | u x => 1)
• Finsupp.isCompl_range_lmapDomain_span 📋 Mathlib.LinearAlgebra.Finsupp.VectorSpace
  {R : Type u_1} {ι : Type u_3} [Semiring R] {α : Type u_4} {β : Type u_5} {u : α → ι} {v : β → ι} (huv : IsCompl (Set.range u) (Set.range v)) : IsCompl (Finsupp.lmapDomain R R u).range (Submodule.span R (Set.range fun x => fun₀ | v x => 1))
• Module.range_piEquiv 📋 Mathlib.LinearAlgebra.StdBasis
  (ι : Type u_1) (R : Type u_2) (M : Type u_3) [Finite ι] [CommSemiring R] [AddCommMonoid M] [Module R M] (v : ι → M) : ((Module.piEquiv ι R M) v).range = Submodule.span R (Set.range v)
• TensorProduct.range_map_eq_span_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : (TensorProduct.map f g).range = Submodule.span R {t | ∃ m n, f m ⊗ₜ[R] g n = t}
• Ideal.range_finsuppTotal 📋 Mathlib.RingTheory.Ideal.Operations
  {ι : Type u_1} {M : Type u_2} [AddCommGroup M] {R : Type u_3} [CommRing R] [Module R M] (I : Ideal R) {v : ι → M} : (Ideal.finsuppTotal ι M I v).range = I • Submodule.span R (Set.range v)
• LinearMap.range_liftBaseChange 📋 Mathlib.LinearAlgebra.TensorProduct.Lift
  {R : Type u_1} {M : Type u_2} {N : Type u_3} (A : Type u_4) [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module A N] [IsScalarTower R A N] (l : M →ₗ[R] N) : (LinearMap.liftBaseChange A l).range = Submodule.span A ↑l.range
• range_vecMulLinear 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {R : Type u_1} [Semiring R] {m : Type u_3} {n : Type u_4} [Fintype m] (M : Matrix m n R) : M.vecMulLinear.range = Submodule.span R (Set.range M.row)
• Matrix.range_mulVecLin 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {R : Type u_1} [CommSemiring R] {m : Type u_4} {n : Type u_5} [Fintype n] (M : Matrix m n R) : M.mulVecLin.range = Submodule.span R (Set.range M.col)
• Matrix.range_toLin' 📋 Mathlib.LinearAlgebra.Matrix.ToLin
  {R : Type u_1} [CommSemiring R] {m : Type u_4} {n : Type u_5} [DecidableEq n] [Fintype n] (M : Matrix m n R) : (Matrix.toLin' M).range = Submodule.span R (Set.range M.col)
• Module.Basis.ofIsLocalizedModule_span 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} (Rₛ : Type u_2) [CommSemiring R] (S : Submonoid R) [CommSemiring Rₛ] [Algebra R Rₛ] [IsLocalization S Rₛ] {M : Type u_3} {Mₛ : Type u_4} [AddCommMonoid M] [Module R M] [AddCommMonoid Mₛ] [Module R Mₛ] [Module Rₛ Mₛ] [IsScalarTower R Rₛ Mₛ] (f : M →ₗ[R] Mₛ) [IsLocalizedModule S f] {ι : Type u_5} (b : Module.Basis ι R M) : Submodule.span R (Set.range ⇑(Module.Basis.ofIsLocalizedModule Rₛ S f b)) = f.range
• Module.Basis.localizationLocalization_span 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} (Rₛ : Type u_2) [CommSemiring R] (S : Submonoid R) [CommSemiring Rₛ] [Algebra R Rₛ] [IsLocalization S Rₛ] {A : Type u_3} [CommSemiring A] [Algebra R A] (Aₛ : Type u_4) [CommSemiring Aₛ] [Algebra A Aₛ] [Algebra Rₛ Aₛ] [Algebra R Aₛ] [IsScalarTower R Rₛ Aₛ] [IsScalarTower R A Aₛ] [IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ] {ι : Type u_5} (b : Module.Basis ι R A) : Submodule.span R (Set.range ⇑(Module.Basis.localizationLocalization Rₛ S Aₛ b)) = (↑(IsScalarTower.toAlgHom R A Aₛ)).range
• LinearMap.range_smulRight_apply_of_surjective 📋 Mathlib.Algebra.Module.LinearMap.DivisionRing
  {R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [AddCommMonoid M] [AddCommMonoid M₁] [Semiring R] [Module R M] [Module R M₁] {f : M →ₗ[R] R} (hf : Function.Surjective ⇑f) (x : M₁) : (f.smulRight x).range = R ∙ x
• LinearMap.range_smulRight_apply 📋 Mathlib.Algebra.Module.LinearMap.DivisionRing
  {R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [AddCommMonoid M] [AddCommMonoid M₁] [DivisionSemiring R] [Module R M] [Module R M₁] {f : M →ₗ[R] R} (hf : f ≠ 0) (x : M₁) : (f.smulRight x).range = R ∙ x
• CharacterModule.intSpanEquivQuotAddOrderOf_symm_apply_coe 📋 Mathlib.Algebra.Module.CharacterModule
  {A : Type uA} [AddCommGroup A] (a : A) (a✝ : ℤ ⧸ Ideal.span {↑(addOrderOf a)}) : ↑((CharacterModule.intSpanEquivQuotAddOrderOf a).symm a✝) = ↑((LinearMap.toSpanSingleton ℤ A a).quotKerEquivRange (((LinearMap.toSpanSingleton ℤ A a).ker.quotEquivOfEq (Ideal.span {↑(addOrderOf a)}) ⋯).symm a✝))
• CharacterModule.intSpanEquivQuotAddOrderOf_apply 📋 Mathlib.Algebra.Module.CharacterModule
  {A : Type uA} [AddCommGroup A] (a : A) (x : ↥(ℤ ∙ a)) : (CharacterModule.intSpanEquivQuotAddOrderOf a) x = ((LinearMap.toSpanSingleton ℤ A a).ker.quotEquivOfEq (Ideal.span {↑(addOrderOf a)}) ⋯) ((LinearMap.toSpanSingleton ℤ A a).quotKerEquivRange.symm ((LinearEquiv.ofEq (ℤ ∙ a) (LinearMap.toSpanSingleton ℤ A a).range ⋯) x))
• Matrix.isRepresentation.toEnd_exists_mem_ideal 📋 Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
  {ι : Type u_1} [Fintype ι] {M : Type u_2} [AddCommGroup M] (R : Type u_3) [CommRing R] [Module R M] (b : ι → M) [DecidableEq ι] (hb : Submodule.span R (Set.range b) = ⊤) (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) : ∃ M_1, (Matrix.isRepresentation.toEnd R b hb) M_1 = f ∧ ∀ (i j : ι), ↑M_1 i j ∈ I
• Module.Relations.Solution.range_π 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] {relations : Module.Relations A} {M : Type v} [AddCommGroup M] [Module A M] (solution : relations.Solution M) : solution.π.range = Submodule.span A (Set.range solution.var)
• Module.Relations.range_map 📋 Mathlib.Algebra.Module.Presentation.Basic
  {A : Type u} [Ring A] (relations : Module.Relations A) : relations.map.range = Submodule.span A (Set.range relations.relation)
• 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))
• ContinuousLinearMap.range_smulRight_apply 📋 Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic
  {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] {R : Type u_9} [DivisionSemiring R] [Module R M₁] [Module R M₂] [TopologicalSpace R] [ContinuousSMul R M₂] {f : M₁ →L[R] R} (hf : f ≠ 0) (x : M₂) : (↑(f.smulRight x)).range = R ∙ x
• LieModule.range_traceForm_le_span_weight 📋 Mathlib.Algebra.Lie.TraceForm
  (K : Type u_2) (L : Type u_3) (M : Type u_4) [LieRing L] [AddCommGroup M] [LieRingModule L M] [Field K] [LieAlgebra K L] [Module K M] [LieModule K L M] [FiniteDimensional K M] [LieRing.IsNilpotent L] [LieModule.LinearWeights K L M] [LieModule.IsTriangularizable K L M] : LinearMap.range (LieModule.traceForm K L M) ≤ Submodule.span K (Set.range (LieModule.Weight.toLinear K L M))
• LinearMap.IsPerfPair.restrictScalars_of_field 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {K : Type u_1} {L : Type u_2} {M : Type u_3} {N : Type u_4} [Field K] [Field L] [Algebra K L] [AddCommGroup M] [AddCommGroup N] [Module L M] [Module L N] [Module K M] [Module K N] [IsScalarTower K L M] (p : M →ₗ[L] N →ₗ[L] L) [p.IsPerfPair] {M' : Type u_5} {N' : Type u_6} [AddCommGroup M'] [AddCommGroup N'] [Module K M'] [Module K N'] [IsScalarTower K L N] (i : M' →ₗ[K] M) (j : N' →ₗ[K] N) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hij : p.IsPerfectCompl (Submodule.span L ↑i.range) (Submodule.span L ↑j.range)) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap K L).range) : (i.restrictScalarsRange₂ j (Algebra.linearMap K L) ⋯ p hp).IsPerfPair
• LinearMap.IsPerfPair.restrictScalars 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : M →ₗ[R] N →ₗ[R] R) [p.IsPerfPair] {S : Type u_4} {M' : Type u_5} {N' : Type u_6} [CommRing S] [IsDomain S] [Algebra S R] [Module S M] [Module S N] [IsScalarTower S R M] [IsScalarTower S R N] [Module.IsTorsionFree S R] [Nontrivial R] [AddCommGroup M'] [Module S M'] [AddCommGroup N'] [Module S N'] (i : M' →ₗ[S] M) (j : N' →ₗ[S] N) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hM : Submodule.span R ↑i.range = ⊤) (hN : Submodule.span R ↑j.range = ⊤) (h₁ : ∀ (g : Module.Dual S N'), ∃ m, ↑S (p.toPerfPair (i m)) ∘ₗ j = Algebra.linearMap S R ∘ₗ g) (h₂ : ∀ (g : Module.Dual S M'), ∃ n, ↑S (p.flip.toPerfPair (j n)) ∘ₗ i = Algebra.linearMap S R ∘ₗ g) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap S R).range) : (i.restrictScalarsRange₂ j (Algebra.linearMap S R) ⋯ p hp).IsPerfPair
• RootPairing.root'_apply_apply_mem_of_mem_span 📋 Mathlib.LinearAlgebra.RootSystem.IsValuedIn
  {ι : Type u_1} {R : Type u_2} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (S : Type u_6) [CommRing S] [Algebra S R] [Module S N] [IsScalarTower S R N] [P.IsValuedIn S] {x : N} (hx : x ∈ Submodule.span S (Set.range ⇑P.coroot)) (i : ι) : (P.root' i) x ∈ (Algebra.linearMap S R).range
• Submodule.span_range_eq_top_of_injective_of_rank_le 📋 Mathlib.Algebra.Module.Lattice
  {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] {M N : Type u} [IsDomain R] [IsFractionRing R K] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module K N] [IsScalarTower R K N] [Module.Finite K N] {f : M →ₗ[R] N} (hf : Function.Injective ⇑f) (h : Module.rank K N ≤ Module.rank R M) : Submodule.span K ↑f.range = ⊤
• Module.Presentation.cokernel 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : Module.Presentation A (M₂ ⧸ f.range)
• Module.Presentation.cokernelSolution.isPresentation 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : (pres₂.cokernelSolution data).IsPresentation
• Module.Presentation.cokernelSolution.isPresentationCore 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : (pres₂.cokernelSolution data).IsPresentationCore
• Module.Presentation.cokernel_G 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : (pres₂.cokernel data hg₁).G = pres₂.G
• Module.Presentation.cokernel_R 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) : (pres₂.cokernel data hg₁).R = (pres₂.R ⊕ ι)
• Module.Presentation.cokernel_relation 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (x✝ : pres₂.R ⊕ ι) : (pres₂.cokernel data hg₁).relation x✝ = match x✝ with | Sum.inl r => pres₂.relation r | Sum.inr i => data.lift i
• Module.Presentation.ofExact_var 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] [AddCommGroup M₃] [Module A M₃] {f : M₁ →ₗ[A] M₂} {g : M₂ →ₗ[A] M₃} (pres₂ : Module.Presentation A M₂) {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (g✝ : (pres₂.cokernel data hg₁).G) : (pres₂.ofExact data hfg hg hg₁).var g✝ = g (pres₂.var g✝)
• Module.Presentation.cokernel_var 📋 Mathlib.Algebra.Module.Presentation.Cokernel
  {A : Type u} [Ring A] {M₁ : Type v₁} {M₂ : Type v₂} [AddCommGroup M₁] [Module A M₁] [AddCommGroup M₂] [Module A M₂] (pres₂ : Module.Presentation A M₂) {f : M₁ →ₗ[A] M₂} {ι : Type w₁} {g₁ : ι → M₁} (data : pres₂.CokernelData f g₁) (hg₁ : Submodule.span A (Set.range g₁) = ⊤) (g : (pres₂.cokernelRelations data).G) : (pres₂.cokernel data hg₁).var g = Submodule.Quotient.mk (pres₂.var g)
• PiTensorProduct.map_range_eq_span_tprod 📋 Mathlib.LinearAlgebra.PiTensorProduct.Basic
  {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ι → Type u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ι → Type u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) : (PiTensorProduct.map f).range = Submodule.span R {t_1 | ∃ m, (⨂ₜ[R] (i : ι), (f i) (m i)) = t_1}
• ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply 📋 Mathlib.Analysis.Normed.Module.Span
  (𝕜 : Type u_1) {E : Type u_2} [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (h : x ≠ 0) (a✝ : ↥(𝕜 ∙ x)) : (ContinuousLinearEquiv.toSpanNonzeroSingleton 𝕜 x h).symm a✝ = (LinearEquiv.ofInjective (LinearMap.toSpanSingleton 𝕜 E x) ⋯).symm ((LinearEquiv.ofEq (𝕜 ∙ x) (LinearMap.toSpanSingleton 𝕜 E x).range ⋯) a✝)
• ZLattice.comap_span_top 📋 Mathlib.Algebra.Module.ZLattice.Basic
  (K : Type u_1) [NormedField K] {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace K E] [NormedAddCommGroup F] [NormedSpace K F] (L : Submodule ℤ E) (hL : Submodule.span K ↑L = ⊤) {e : F →ₗ[K] E} (he : ↑L ⊆ ↑e.range) : Submodule.span K ↑(ZLattice.comap K L e) = ⊤
• IsIntegralClosure.range_le_span_dualBasis 📋 Mathlib.RingTheory.DedekindDomain.IntegralClosure
  {A : Type u_1} {K : Type u_2} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_3} [Field L] (C : Type u_4) [CommRing C] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [Algebra C L] [IsIntegralClosure C A L] [Algebra A C] [IsScalarTower A C L] [FiniteDimensional K L] [Algebra.IsSeparable K L] {ι : Type u_5} [Finite ι] [DecidableEq ι] (b : Module.Basis ι K L) (hb_int : ∀ (i : ι), IsIntegral A (b i)) [IsIntegrallyClosed A] : (↑A (Algebra.linearMap C L)).range ≤ Submodule.span A (Set.range ⇑((Algebra.traceForm K L).dualBasis ⋯ b))
• SchauderBasis.range_proj_eq_span 📋 Mathlib.Analysis.Normed.Module.Bases
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace 𝕜 X] (b : SchauderBasis 𝕜 X) (n : ℕ) : (↑(b.proj n)).range = Submodule.span 𝕜 (↑b '' ↑(Finset.range n))
• GeneralSchauderBasis.range_proj_eq_span 📋 Mathlib.Analysis.Normed.Module.Bases
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {X : Type u_2} [NormedAddCommGroup X] [NormedSpace 𝕜 X] {β : Type u_3} {L : SummationFilter β} (b : GeneralSchauderBasis β 𝕜 X L) (A : Finset β) : (↑(b.proj A)).range = Submodule.span 𝕜 (↑b '' ↑A)
• range_mvfderiv_subtypeVal 📋 Mathlib.Geometry.Manifold.Instances.Sphere
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [Fact (Module.finrank ℝ E = n + 1)] (v : ↑(Metric.sphere 0 1)) : (↑(d% Subtype.val v)).range = (ℝ ∙ ↑v)ᗮ
• range_mfderiv_coe_sphere 📋 Mathlib.Geometry.Manifold.Instances.Sphere
  {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {n : ℕ} [Fact (Module.finrank ℝ E = n + 1)] (v : ↑(Metric.sphere 0 1)) : (↑(mfderiv% Subtype.val v)).range = (ℝ ∙ ↑v)ᗮ
• RootPairing.GeckConstruction.span_range_h_le_range_diagonal 📋 Mathlib.LinearAlgebra.RootSystem.GeckConstruction.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} [P.IsCrystallographic] {b : P.Base} [DecidableEq ι] : Submodule.span R (Set.range RootPairing.GeckConstruction.h) ≤ (Matrix.diagonalLinearMap (↥b.support ⊕ ι) R R).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 9d63ad7