Loogle!

Result

Found 54 declarations mentioning Subalgebra.map.

Subalgebra.map 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R A) : Subalgebra R B
Subalgebra.map_id 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (S : Subalgebra R A) : Subalgebra.map (AlgHom.id R A) S = S
Subalgebra.map_injective 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {f : A →ₐ[R] B} (hf : Function.Injective ⇑f) : Function.Injective (Subalgebra.map f)
Subalgebra.comap_map_eq_self_of_injective 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {f : A →ₐ[R] B} (hf : Function.Injective ⇑f) (S : Subalgebra R A) : Subalgebra.comap f (Subalgebra.map f S) = S
AlgHom.range_comp 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} {C : Type w'} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] (f : A →ₐ[R] B) (g : B →ₐ[R] C) : (g.comp f).range = Subalgebra.map g f.range
Subalgebra.coe_map 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R A) : ↑(Subalgebra.map f S) = ⇑f '' ↑S
Subalgebra.gc_map_comap 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : GaloisConnection (Subalgebra.map f) (Subalgebra.comap f)
Subalgebra.map_map 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} {C : Type w'} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] (S : Subalgebra R A) (g : B →ₐ[R] C) (f : A →ₐ[R] B) : Subalgebra.map g (Subalgebra.map f S) = Subalgebra.map (g.comp f) S
Subalgebra.map_toSubsemiring 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R A) : (Subalgebra.map f S).toSubsemiring = Subsemiring.map (↑f) S.toSubsemiring
Subalgebra.mem_map 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {S : Subalgebra R A} {f : A →ₐ[R] B} {y : B} : y ∈ Subalgebra.map f S ↔ ∃ x ∈ S, f x = y
Subalgebra.map_mono 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {S₁ S₂ : Subalgebra R A} {f : A →ₐ[R] B} : S₁ ≤ S₂ → Subalgebra.map f S₁ ≤ Subalgebra.map f S₂
Subalgebra.map_le 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {S : Subalgebra R A} {f : A →ₐ[R] B} {U : Subalgebra R B} : Subalgebra.map f S ≤ U ↔ S ≤ Subalgebra.comap f U
AlgHom.subalgebraMap 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) : ↥S →ₐ[R] ↥(Subalgebra.map f S)
Subalgebra.range_comp_val 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) : (f.comp S.val).range = Subalgebra.map f S
Subalgebra.equivMapOfInjective 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) : ↥S ≃ₐ[R] ↥(Subalgebra.map f S)
AlgEquiv.subalgebraMap 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (e : A ≃ₐ[R] B) (S : Subalgebra R A) : ↥S ≃ₐ[R] ↥(Subalgebra.map (↑e) S)
AlgHom.subalgebraMap_surjective 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) : Function.Surjective ⇑(AlgHom.subalgebraMap S f)
AlgHom.subalgebraMap_coe_apply 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {S : Subalgebra R A} (f : A →ₐ[R] B) (x : ↥S) : ↑((AlgHom.subalgebraMap S f) x) = f ↑x
Subalgebra.coe_equivMapOfInjective_apply 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (x : ↥S) : ↑((S.equivMapOfInjective f hf) x) = f ↑x
AlgEquiv.subalgebraMap_apply_coe 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (e : A ≃ₐ[R] B) (S : Subalgebra R A) (x : ↑↑S.toAddSubmonoid) : ↑((e.subalgebraMap S) x) = e ↑x
Subalgebra.map_toSubmodule 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {S : Subalgebra R A} {f : A →ₐ[R] B} : Subalgebra.toSubmodule (Subalgebra.map f S) = Submodule.map f.toLinearMap (Subalgebra.toSubmodule S)
AlgEquiv.subalgebraMap_symm_apply_coe 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (e : A ≃ₐ[R] B) (S : Subalgebra R A) (y : ↑(⇑↑e.toRingEquiv.toAddEquiv '' ↑S.toAddSubmonoid)) : ↑((e.subalgebraMap S).symm y) = (↑↑e).symm ↑y
AlgHom.map_adjoin 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (s : Set A) : Subalgebra.map φ (Algebra.adjoin R s) = Algebra.adjoin R (⇑φ '' s)
Algebra.adjoin_image 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
(R : Type uR) {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (s : Set A) : Algebra.adjoin R (⇑f '' s) = Subalgebra.map f (Algebra.adjoin R s)
Subalgebra.map_comap_eq_self_of_surjective 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {f : A →ₐ[R] B} (hf : Function.Surjective ⇑f) (S : Subalgebra R B) : Subalgebra.map f (Subalgebra.comap f S) = S
AlgHom.map_adjoin_singleton 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (e : A →ₐ[R] B) (x : A) : Subalgebra.map e (Algebra.adjoin R {x}) = Algebra.adjoin R {e x}
Subalgebra.map_comap_eq 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R B) : Subalgebra.map f (Subalgebra.comap f S) = S ⊓ f.range
Subalgebra.map_comap_eq_self 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {f : A →ₐ[R] B} {S : Subalgebra R B} (h : S ≤ f.range) : Subalgebra.map f (Subalgebra.comap f S) = S
Algebra.map_iInf 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {ι : Sort u_1} [Nonempty ι] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (s : ι → Subalgebra R A) : Subalgebra.map f (iInf s) = ⨅ i, Subalgebra.map f (s i)
Algebra.map_sup 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S T : Subalgebra R A) : Subalgebra.map f (S ⊔ T) = Subalgebra.map f S ⊔ Subalgebra.map f T
Subalgebra.comap_map_eq_self 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B] {f : A →ₐ[R] B} {S : Subalgebra R A} (h : ⇑f ⁻¹' {0} ⊆ ↑S) : Subalgebra.comap f (Subalgebra.map f S) = S
Algebra.map_top 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : Subalgebra.map f ⊤ = f.range
Algebra.map_inf 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (S T : Subalgebra R A) : Subalgebra.map f (S ⊓ T) = Subalgebra.map f S ⊓ Subalgebra.map f T
Subalgebra.comap_map_eq 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Ring A] [Algebra R A] [Ring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R A) : Subalgebra.comap f (Subalgebra.map f S) = S ⊔ Algebra.adjoin R (⇑f ⁻¹' {0})
Algebra.map_bot 📋 Mathlib.Algebra.Algebra.Subalgebra.Lattice
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : Subalgebra.map f ⊥ = ⊥
Subalgebra.centralizer_coe_map_includeRight_eq_center_tensorProduct 📋 Mathlib.Algebra.Algebra.Subalgebra.Centralizer
(R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [Algebra R A] (B : Type u_3) [Semiring B] [Algebra R B] (S : Subalgebra R B) [Module.Free R A] : Subalgebra.centralizer R ↑(Subalgebra.map Algebra.TensorProduct.includeRight S) = (Algebra.TensorProduct.map (AlgHom.id R A) (Subalgebra.centralizer R ↑S).val).range
Subalgebra.centralizer_coe_map_includeLeft_eq_center_tensorProduct 📋 Mathlib.Algebra.Algebra.Subalgebra.Centralizer
(R : Type u_1) [CommSemiring R] (A : Type u_2) [Semiring A] [Algebra R A] (B : Type u_3) [Semiring B] [Algebra R B] (S : Subalgebra R A) [Module.Free R B] : Subalgebra.centralizer R ↑(Subalgebra.map Algebra.TensorProduct.includeLeft S) = (Algebra.TensorProduct.map (Subalgebra.centralizer R ↑S).val (AlgHom.id R B)).range
Algebra.adjoin_algebraMap 📋 Mathlib.RingTheory.Adjoin.Basic
(R : Type uR) {S : Type uS} (A : Type uA) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] (s : Set S) : Algebra.adjoin R (⇑(algebraMap S A) '' s) = Subalgebra.map (IsScalarTower.toAlgHom R S A) (Algebra.adjoin R s)
Subalgebra.FG.map 📋 Mathlib.RingTheory.Adjoin.FG
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {S : Subalgebra R A} (f : A →ₐ[R] B) (hs : S.FG) : (Subalgebra.map f S).FG
Subalgebra.fg_of_fg_map 📋 Mathlib.RingTheory.Adjoin.FG
{R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (S : Subalgebra R A) (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (hs : (Subalgebra.map f S).FG) : S.FG
Algebra.adjoin_restrictScalars 📋 Mathlib.RingTheory.Adjoin.Tower
(C : Type u_1) (D : Type u_2) (E : Type u_3) [CommSemiring C] [CommSemiring D] [CommSemiring E] [Algebra C D] [Algebra C E] [Algebra D E] [IsScalarTower C D E] (S : Set E) : Subalgebra.restrictScalars C (Algebra.adjoin D S) = Subalgebra.restrictScalars C (Algebra.adjoin (↥(Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤)) S)
StarSubalgebra.map_toSubalgebra 📋 Mathlib.Algebra.Star.Subalgebra
{R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [StarRing R] [Semiring A] [StarRing A] [Algebra R A] [StarModule R A] [Semiring B] [StarRing B] [Algebra R B] [StarModule R B] {S : StarSubalgebra R A} {f : A →⋆ₐ[R] B} : (StarSubalgebra.map f S).toSubalgebra = Subalgebra.map f.toAlgHom S.toSubalgebra
Algebra.EssFiniteType.aux 📋 Mathlib.RingTheory.EssentialFiniteness
(R : Type u_1) (S : Type u_2) (T : Type u_3) [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (σ : Subalgebra R S) (hσ : ∀ (s : S), ∃ t ∈ σ, IsUnit t ∧ s * t ∈ σ) (τ : Set T) (t : T) (ht : t ∈ Algebra.adjoin S τ) : ∃ s ∈ σ, IsUnit s ∧ s • t ∈ Subalgebra.map (IsScalarTower.toAlgHom R S T) σ ⊔ Algebra.adjoin R τ
integralClosure_map_algEquiv 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
{R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) : Subalgebra.map (↑f) (integralClosure R A) = integralClosure R S
Algebra.TensorProduct.includeRight_map_center_le 📋 Mathlib.Algebra.Central.TensorProduct
(K : Type u_1) (B : Type u_2) (C : Type u_3) [CommSemiring K] [Semiring B] [Semiring C] [Algebra K B] [Algebra K C] : Subalgebra.map Algebra.TensorProduct.includeRight (Subalgebra.center K C) ≤ Subalgebra.center K (TensorProduct K B C)
Algebra.TensorProduct.includeLeft_map_center_le 📋 Mathlib.Algebra.Central.TensorProduct
(K : Type u_1) (B : Type u_2) (C : Type u_3) [CommSemiring K] [Semiring B] [Semiring C] [Algebra K B] [Algebra K C] : Subalgebra.map Algebra.TensorProduct.includeLeft (Subalgebra.center K B) ≤ Subalgebra.center K (TensorProduct K B C)
IntermediateField.toSubalgebra_map 📋 Mathlib.FieldTheory.IntermediateField.Basic
{K : Type u_1} {L : Type u_2} {L' : Type u_3} [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L'] (S : IntermediateField K L) (f : L →ₐ[K] L') : (IntermediateField.map f S).toSubalgebra = Subalgebra.map f S.toSubalgebra
Subalgebra.topologicalClosure_map 📋 Mathlib.Topology.Algebra.Algebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [IsTopologicalSemiring A] [IsTopologicalSemiring B] (f : A →A[R] B) (s : Subalgebra R A) : Subalgebra.map (↑f) s.topologicalClosure ≤ (Subalgebra.map f.toAlgHom s).topologicalClosure
DenseRange.topologicalClosure_map_subalgebra 📋 Mathlib.Topology.Algebra.Algebra
{R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [TopologicalSpace A] {B : Type u_3} [Semiring B] [TopologicalSpace B] [Algebra R A] [Algebra R B] [IsTopologicalSemiring A] [IsTopologicalSemiring B] {f : A →A[R] B} (hf' : DenseRange ⇑f) {s : Subalgebra R A} (hs : s.topologicalClosure = ⊤) : (Subalgebra.map (↑f) s).topologicalClosure = ⊤
polynomialFunctions.eq_1 📋 Mathlib.Topology.ContinuousMap.Polynomial
{R : Type u_1} [CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] (X : Set R) : polynomialFunctions X = Subalgebra.map (Polynomial.toContinuousMapOnAlgHom X) ⊤
RCLike.restrict_toContinuousMap_eq_toContinuousMapStar_restrict 📋 Mathlib.Analysis.RCLike.BoundedContinuous
(𝕜 : Type u_1) (E : Type u_2) [RCLike 𝕜] [PseudoEMetricSpace E] {A : StarSubalgebra 𝕜 (BoundedContinuousFunction E 𝕜)} : Subalgebra.map (BoundedContinuousFunction.toContinuousMapₐ ℝ) (Subalgebra.comap (BoundedContinuousFunction.AlgHom.compLeftContinuousBounded ℝ RCLike.ofRealAm ⋯) (Subalgebra.restrictScalars ℝ A.toSubalgebra)) = Subalgebra.comap (AlgHom.compLeftContinuous ℝ RCLike.ofRealAm ⋯) (Subalgebra.restrictScalars ℝ (StarSubalgebra.map (BoundedContinuousFunction.toContinuousMapStarₐ 𝕜) A).toSubalgebra)
dist_integral_mulExpNegMulSq_comp_le 📋 Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral
{ε : ℝ} {E : Type u_2} [MeasurableSpace E] [PseudoEMetricSpace E] [BorelSpace E] [CompleteSpace E] [SecondCountableTopology E] {P P' : MeasureTheory.Measure E} [MeasureTheory.IsFiniteMeasure P] [MeasureTheory.IsFiniteMeasure P'] (f : BoundedContinuousFunction E ℝ) {A : Subalgebra ℝ (BoundedContinuousFunction E ℝ)} (hA : (Subalgebra.map (BoundedContinuousFunction.toContinuousMapₐ ℝ) A).SeparatesPoints) (heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P') (hε : 0 < ε) : |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (f x) ∂P'| ≤ 6 * √ε
Subalgebra.mulMap_map_comp_eq 📋 Mathlib.LinearAlgebra.TensorProduct.Subalgebra
{R : Type u_1} {S : Type u_2} {T : Type u_3} [CommSemiring R] [CommSemiring S] [Algebra R S] [CommSemiring T] [Algebra R T] (A B : Subalgebra R S) (f : S →ₐ[R] T) : ((Subalgebra.map f A).mulMap (Subalgebra.map f B)).comp (Algebra.TensorProduct.map (AlgHom.subalgebraMap A f) (AlgHom.subalgebraMap B f)) = f.comp (A.mulMap B)
Subalgebra.LinearDisjoint.map 📋 Mathlib.RingTheory.LinearDisjoint
{R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {A B : Subalgebra R S} (H : A.LinearDisjoint B) {T : Type w} [Semiring T] [Algebra R T] (f : S →ₐ[R] T) (hf : Function.Injective ⇑f) : (Subalgebra.map f A).LinearDisjoint (Subalgebra.map f B)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. 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.

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.

This is Loogle revision ff04530 serving mathlib revision 8623f65