Loogle!

Result

Found 612 declarations mentioning RingHom.comp. Of these, only the first 200 are shown.

• RingHom.comp 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} {x✝² : NonAssocSemiring γ} (g : β →+* γ) (f : α →+* β) : α →+* γ
• RingHom.comp_id 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} (f : α →+* β) : f.comp (RingHom.id α) = f
• RingHom.id_comp 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} (f : α →+* β) : (RingHom.id β).comp f = f
• RingHom.mul_def 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {x✝ : NonAssocSemiring α} (f g : α →+* α) : f * g = f.comp g
• RingHom.comp_assoc 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} {x✝² : NonAssocSemiring γ} {δ : Type u_5} {x✝³ : NonAssocSemiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) : (h.comp g).comp f = h.comp (g.comp f)
• RingHom.cancel_left 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} {x✝² : NonAssocSemiring γ} {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
• RingHom.cancel_right 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} {x✝² : NonAssocSemiring γ} {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂
• RingHom.comp_apply 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} {x✝² : NonAssocSemiring γ} (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn) x = hnp (hmn x)
• RingHom.coe_comp 📋 Mathlib.Algebra.Ring.Hom.Defs
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} {x✝ : NonAssocSemiring α} {x✝¹ : NonAssocSemiring β} {x✝² : NonAssocSemiring γ} (hnp : β →+* γ) (hmn : α →+* β) : ⇑(hnp.comp hmn) = ⇑hnp ∘ ⇑hmn
• RingEquiv.ofRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) : R ≃+* S
• RingEquiv.symm_toRingHom_comp_toRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : e.symm.toRingHom.comp e.toRingHom = RingHom.id R
• RingEquiv.toRingHom_comp_symm_toRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : e.toRingHom.comp e.symm.toRingHom = RingHom.id S
• RingEquiv.ofRingHom_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) : (RingEquiv.ofRingHom f g h₁ h₂).symm = RingEquiv.ofRingHom g f h₂ h₁
• RingEquiv.toRingHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂).toRingHom = e₂.toRingHom.comp e₁.toRingHom
• RingEquiv.ofRingHom_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) (a : R) : (RingEquiv.ofRingHom f g h₁ h₂) a = f a
• RingEquiv.ofRingHom_symm_apply 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) (a : S) : (RingEquiv.ofRingHom f g h₁ h₂).symm a = g a
• RingEquiv.coe_ringHom_ofRingHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S) (h₂ : g.comp f = RingHom.id R) : ↑(RingEquiv.ofRingHom f g h₁ h₂) = f
• RingEquiv.comp_symm 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : (↑e).comp ↑e.symm = RingHom.id S
• RingEquiv.symm_comp 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : (↑e.symm).comp ↑e = RingHom.id R
• RingEquiv.ofRingHom_coe_ringHom 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (g : S →+* R) (h₁ : (↑f).comp g = RingHom.id S) (h₂ : g.comp ↑f = RingHom.id R) : RingEquiv.ofRingHom (↑f) g h₁ h₂ = f
• RingEquiv.coe_ringHom_trans 📋 Mathlib.Algebra.Ring.Equiv
  {R : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring S'] (e₁ : R ≃+* S) (e₂ : S ≃+* S') : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁
• RingHomInvPair.comp_eq 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} {inst✝ : Semiring R₁} {inst✝¹ : Semiring R₂} {σ : R₁ →+* R₂} {σ' : outParam (R₂ →+* R₁)} [self : RingHomInvPair σ σ'] : σ'.comp σ = RingHom.id R₁
• RingHomInvPair.comp_eq₂ 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} {inst✝ : Semiring R₁} {inst✝¹ : Semiring R₂} {σ : R₁ →+* R₂} {σ' : outParam (R₂ →+* R₁)} [self : RingHomInvPair σ σ'] : σ.comp σ' = RingHom.id R₂
• RingHomCompTriple.comp_eq 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {inst✝ : Semiring R₁} {inst✝¹ : Semiring R₂} {inst✝² : Semiring R₃} {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : outParam (R₁ →+* R₃)} [self : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] : σ₂₃.comp σ₁₂ = σ₁₃
• RingHomCompTriple.mk 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [Semiring R₁] [Semiring R₂] [Semiring R₃] {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : outParam (R₁ →+* R₃)} (comp_eq : σ₂₃.comp σ₁₂ = σ₁₃) : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
• RingHomInvPair.mk 📋 Mathlib.Algebra.Ring.CompTypeclasses
  {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ : R₁ →+* R₂} {σ' : outParam (R₂ →+* R₁)} (comp_eq : σ'.comp σ = RingHom.id R₁) (comp_eq₂ : σ.comp σ' = RingHom.id R₂) : RingHomInvPair σ σ'
• Algebra.compHom_algebraMap_eq 📋 Mathlib.Algebra.Algebra.Defs
  {R : Type u} {S : Type v} (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R A] (f : S →+* R) : algebraMap S A = (algebraMap R A).comp f
• RingHom.comp_ulift_eq 📋 Mathlib.Algebra.Ring.ULift
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (f : R →+* S) : ULift.ringEquiv.toRingHom.comp ((RingHom.ulift.{u₁, u₂, u_1, u_2} f).comp ULift.ringEquiv.symm.toRingHom) = f
• RingHom.fst_comp_prod 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) : (RingHom.fst S T).comp (f.prod g) = f
• RingHom.snd_comp_prod 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) : (RingHom.snd S T).comp (f.prod g) = g
• RingHom.prod_unique 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {S : Type u_3} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S × T) : ((RingHom.fst S T).comp f).prod ((RingHom.snd S T).comp f) = f
• RingHom.prodMap_def 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') : f.prodMap g = (f.comp (RingHom.fst R S)).prod (g.comp (RingHom.snd R S))
• RingHom.prod_comp_prodMap 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} {T : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] [NonAssocSemiring T] (f : T →+* R) (g : T →+* S) (f' : R →+* R') (g' : S →+* S') : (f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
• RingEquiv.fst_comp_coe_prodComm 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] : (RingHom.fst S R).comp ↑RingEquiv.prodComm = RingHom.snd R S
• RingEquiv.snd_comp_coe_prodComm 📋 Mathlib.Algebra.Ring.Prod
  {R : Type u_1} {S : Type u_3} [NonAssocSemiring R] [NonAssocSemiring S] : (RingHom.snd S R).comp ↑RingEquiv.prodComm = RingHom.fst R S
• RingHom.map_rangeS 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) : Subsemiring.map g f.rangeS = (g.comp f).rangeS
• Subsemiring.comap_comap 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring T) (g : S →+* T) (f : R →+* S) : Subsemiring.comap f (Subsemiring.comap g s) = Subsemiring.comap (g.comp f) s
• Subsemiring.map_map 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring R) (g : S →+* T) (f : R →+* S) : Subsemiring.map g (Subsemiring.map f s) = Subsemiring.map (g.comp f) s
• RingHom.domRestrict_comp_rangeSRestrict 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) : (g.domRestrict f.rangeS).comp f.rangeSRestrict = g.comp f
• RingHom.domRestrict_comp_codRestrict 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) (p : Subsemiring S) (h : ∀ (c : R), f c ∈ p) : (g.domRestrict p).comp (f.codRestrict p h) = g.comp f
• RingHom.comp_restrict 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) : (SubsemiringClass.subtype s).comp (f.restrict s' s h) = f.comp (SubsemiringClass.subtype s')
• RingHom.map_range 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocRing R] [NonAssocRing S] [NonAssocRing T] (g : S →+* T) (f : R →+* S) : Subring.map g f.range = (g.comp f).range
• Subring.comap_comap 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocRing R] [NonAssocRing S] [NonAssocRing T] (s : Subring T) (g : S →+* T) (f : R →+* S) : Subring.comap f (Subring.comap g s) = Subring.comap (g.comp f) s
• Subring.map_map 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocRing R] [NonAssocRing S] [NonAssocRing T] (s : Subring R) (g : S →+* T) (f : R →+* S) : Subring.map g (Subring.map f s) = Subring.map (g.comp f) s
• RingHom.domRestrict_comp_rangeRestrict 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocRing R] [NonAssocRing S] [NonAssocRing T] (g : S →+* T) (f : R →+* S) : (g.domRestrict f.range).comp f.rangeRestrict = g.comp f
• AlgHom.comp_algebraMap 📋 Mathlib.Algebra.Algebra.Hom
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : (↑φ).comp (algebraMap R A) = algebraMap R B
• AlgHom.comp_toRingHom 📋 Mathlib.Algebra.Algebra.Hom
  {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂
• AlgEquiv.toRingHom_trans 📋 Mathlib.Algebra.Algebra.Equiv
  {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂} {A₃ : Type uA₃} [CommSemiring R] [Semiring A₁] [Semiring A₂] [Semiring A₃] [Algebra R A₁] [Algebra R A₂] [Algebra R A₃] (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : ↑(e₁.trans e₂) = (↑e₂).comp ↑e₁
• IsScalarTower.algebraMap_eq 📋 Mathlib.Algebra.Algebra.Tower
  (R : Type u) (S : Type v) (A : Type w) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] : algebraMap R A = (algebraMap S A).comp (algebraMap R S)
• IsScalarTower.of_algebraMap_eq' 📋 Mathlib.Algebra.Algebra.Tower
  {R : Type u} {S : Type v} {A : Type w} [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] (h : algebraMap R A = (algebraMap S A).comp (algebraMap R S)) : IsScalarTower R S A
• AlgHom.comp_algebraMap_of_tower 📋 Mathlib.Algebra.Algebra.Tower
  (R : Type u) {S : Type v} {A : Type w} {B : Type u₁} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R S] [Algebra S A] [Algebra S B] [Algebra R A] [Algebra R B] [IsScalarTower R S A] [IsScalarTower R S B] (f : A →ₐ[S] B) : (↑f).comp (algebraMap R A) = algebraMap R B
• Algebra.rank_eq_of_equiv_equiv 📋 Mathlib.LinearAlgebra.Dimension.Basic
  {R : Type w} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {R' : Type w'} [CommSemiring R'] {S' : Type v} [Semiring S'] [Algebra R' S'] (i : R ≃+* R') (j : S ≃+* S') (hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)) : Module.rank R S = Module.rank R' S'
• Algebra.lift_rank_eq_of_equiv_equiv 📋 Mathlib.LinearAlgebra.Dimension.Basic
  {R : Type w} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {R' : Type w'} {S' : Type v'} [CommSemiring R'] [Semiring S'] [Algebra R' S'] (i : R ≃+* R') (j : S ≃+* S') (hc : (algebraMap R' S').comp i.toRingHom = j.toRingHom.comp (algebraMap R S)) : Cardinal.lift.{v', v} (Module.rank R S) = Cardinal.lift.{v, v'} (Module.rank R' S')
• Algebra.rank_le_of_injective_injective 📋 Mathlib.LinearAlgebra.Dimension.Basic
  {R : Type w} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {R' : Type w'} [CommSemiring R'] {S' : Type v} [Semiring S'] [Algebra R' S'] (i : R' →+* R) (j : S →+* S') (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hc : (j.comp (algebraMap R S)).comp i = algebraMap R' S') : Module.rank R S ≤ Module.rank R' S'
• Algebra.rank_le_of_surjective_injective 📋 Mathlib.LinearAlgebra.Dimension.Basic
  {R : Type w} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {R' : Type w'} [CommSemiring R'] {S' : Type v} [Semiring S'] [Algebra R' S'] (i : R →+* R') (j : S →+* S') (hi : Function.Surjective ⇑i) (hj : Function.Injective ⇑j) (hc : (algebraMap R' S').comp i = j.comp (algebraMap R S)) : Module.rank R S ≤ Module.rank R' S'
• Algebra.lift_rank_le_of_injective_injective 📋 Mathlib.LinearAlgebra.Dimension.Basic
  {R : Type w} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {R' : Type w'} {S' : Type v'} [CommSemiring R'] [Semiring S'] [Algebra R' S'] (i : R' →+* R) (j : S →+* S') (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hc : (j.comp (algebraMap R S)).comp i = algebraMap R' S') : Cardinal.lift.{v', v} (Module.rank R S) ≤ Cardinal.lift.{v, v'} (Module.rank R' S')
• Algebra.lift_rank_le_of_surjective_injective 📋 Mathlib.LinearAlgebra.Dimension.Basic
  {R : Type w} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {R' : Type w'} {S' : Type v'} [CommSemiring R'] [Semiring S'] [Algebra R' S'] (i : R →+* R') (j : S →+* S') (hi : Function.Surjective ⇑i) (hj : Function.Injective ⇑j) (hc : (algebraMap R' S').comp i = j.comp (algebraMap R S)) : Cardinal.lift.{v', v} (Module.rank R S) ≤ Cardinal.lift.{v, v'} (Module.rank R' S')
• Algebra.finrank_eq_of_equiv_equiv 📋 Mathlib.LinearAlgebra.Dimension.Finrank
  {R₀ : Type u_1} {S₀ : Type u_2} [CommSemiring R₀] [Semiring S₀] [Algebra R₀ S₀] {R₁ : Type u_3} {S₁ : Type u_4} [CommSemiring R₁] [Semiring S₁] [Algebra R₁ S₁] (i : R₀ ≃+* R₁) (j : S₀ ≃+* S₁) (hc : (algebraMap R₁ S₁).comp i.toRingHom = j.toRingHom.comp (algebraMap R₀ S₀)) : Module.finrank R₀ S₀ = Module.finrank R₁ S₁
• RingCon.comap_ringHomComp 📋 Mathlib.RingTheory.Congruence.Defs
  {R : Type u_5} {R' : Type u_6} {R'' : Type u_7} [NonAssocSemiring R] [NonAssocSemiring R'] [NonAssocSemiring R''] (J : RingCon R) (g : R' →+* R) (f : R'' →+* R') : J.comap (g.comp f) = (J.comap g).comap f
• Ideal.Quotient.lift_comp_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Defs
  {R : Type u} [Ring R] (I : Ideal R) {S : Type v} [I.IsTwoSided] [Semiring S] (f : R →+* S) (H : ∀ a ∈ I, f a = 0) : (Ideal.Quotient.lift I f H).comp (Ideal.Quotient.mk I) = f
• Ideal.Quotient.factor_comp_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Defs
  {R : Type u} [Ring R] {S T : Ideal R} [S.IsTwoSided] [T.IsTwoSided] (H : S ≤ T) : (Ideal.Quotient.factor H).comp (Ideal.Quotient.mk S) = Ideal.Quotient.mk T
• Ideal.Quotient.ringHom_ext 📋 Mathlib.RingTheory.Ideal.Quotient.Defs
  {R : Type u} [Ring R] {I : Ideal R} {S : Type v} [I.IsTwoSided] [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (Ideal.Quotient.mk I) = g.comp (Ideal.Quotient.mk I)) : f = g
• Ideal.Quotient.ringHom_ext_iff 📋 Mathlib.RingTheory.Ideal.Quotient.Defs
  {R : Type u} [Ring R] {I : Ideal R} {S : Type v} [I.IsTwoSided] [NonAssocSemiring S] {f g : R ⧸ I →+* S} : f = g ↔ f.comp (Ideal.Quotient.mk I) = g.comp (Ideal.Quotient.mk I)
• Ideal.Quotient.factor_comp 📋 Mathlib.RingTheory.Ideal.Quotient.Defs
  {R : Type u} [Ring R] {S T U : Ideal R} [S.IsTwoSided] [T.IsTwoSided] [U.IsTwoSided] (H1 : S ≤ T) (H2 : T ≤ U) : (Ideal.Quotient.factor H2).comp (Ideal.Quotient.factor H1) = Ideal.Quotient.factor ⋯
• RingHom.Finite.of_comp_finite 📋 Mathlib.RingTheory.Finiteness.Basic
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {f : A →+* B} {g : B →+* C} (h : (g.comp f).Finite) : g.Finite
• RingHom.Finite.comp 📋 Mathlib.RingTheory.Finiteness.Basic
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {g : B →+* C} {f : A →+* B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite
• Module.Finite.of_equiv_equiv 📋 Mathlib.RingTheory.Finiteness.Basic
  {A₁ : Type u_5} {B₁ : Type u_6} {A₂ : Type u_7} {B₂ : Type u_8} [CommSemiring A₁] [CommSemiring B₁] [CommSemiring A₂] [Semiring B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂) (he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)) [Module.Finite A₁ B₁] : Module.Finite A₂ B₂
• MonoidAlgebra.ringHom_ext' 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {S : Type u_2} {M : Type u_4} [Semiring R] [MulOneClass M] [Semiring S] {f g : MonoidAlgebra R M →+* S} (h₁ : f.comp MonoidAlgebra.singleOneRingHom = g.comp MonoidAlgebra.singleOneRingHom) (h_of : (↑f).comp (MonoidAlgebra.of R M) = (↑g).comp (MonoidAlgebra.of R M)) : f = g
• MonoidAlgebra.ringHom_ext'_iff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {S : Type u_2} {M : Type u_4} [Semiring R] [MulOneClass M] [Semiring S] {f g : MonoidAlgebra R M →+* S} : f = g ↔ f.comp MonoidAlgebra.singleOneRingHom = g.comp MonoidAlgebra.singleOneRingHom ∧ (↑f).comp (MonoidAlgebra.of R M) = (↑g).comp (MonoidAlgebra.of R M)
• AddMonoidAlgebra.ringHom_ext' 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {S : Type u_2} {M : Type u_4} [Semiring R] [Semiring S] [AddMonoid M] {f g : AddMonoidAlgebra R M →+* S} (h₁ : f.comp AddMonoidAlgebra.singleZeroRingHom = g.comp AddMonoidAlgebra.singleZeroRingHom) (h_of : (↑f).comp (AddMonoidAlgebra.of R M) = (↑g).comp (AddMonoidAlgebra.of R M)) : f = g
• AddMonoidAlgebra.ringHom_ext'_iff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {S : Type u_2} {M : Type u_4} [Semiring R] [Semiring S] [AddMonoid M] {f g : AddMonoidAlgebra R M →+* S} : f = g ↔ f.comp AddMonoidAlgebra.singleZeroRingHom = g.comp AddMonoidAlgebra.singleZeroRingHom ∧ (↑f).comp (AddMonoidAlgebra.of R M) = (↑g).comp (AddMonoidAlgebra.of R M)
• AddMonoidAlgebra.mapRingHom_comp 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {M : Type u_6} [Semiring R] [Semiring S] [Semiring T] [AddMonoid M] (f : S →+* T) (g : R →+* S) : AddMonoidAlgebra.mapRingHom M (f.comp g) = (AddMonoidAlgebra.mapRingHom M f).comp (AddMonoidAlgebra.mapRingHom M g)
• MonoidAlgebra.mapRangeRingHom_comp 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {M : Type u_6} [Semiring R] [Semiring S] [Semiring T] [Monoid M] (f : S →+* T) (g : R →+* S) : MonoidAlgebra.mapRingHom M (f.comp g) = (MonoidAlgebra.mapRingHom M f).comp (MonoidAlgebra.mapRingHom M g)
• MonoidAlgebra.mapRingHom_comp 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {M : Type u_6} [Semiring R] [Semiring S] [Semiring T] [Monoid M] (f : S →+* T) (g : R →+* S) : MonoidAlgebra.mapRingHom M (f.comp g) = (MonoidAlgebra.mapRingHom M f).comp (MonoidAlgebra.mapRingHom M g)
• AddMonoidAlgebra.mapDomainRingHom_comp 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} {O : Type u_8} [Semiring R] [AddMonoid M] [AddMonoid N] [AddMonoid O] (f : N →+ O) (g : M →+ N) : AddMonoidAlgebra.mapDomainRingHom R (f.comp g) = (AddMonoidAlgebra.mapDomainRingHom R f).comp (AddMonoidAlgebra.mapDomainRingHom R g)
• MonoidAlgebra.mapDomainRingHom_comp 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} {O : Type u_8} [Semiring R] [Monoid M] [Monoid N] [Monoid O] (f : N →* O) (g : M →* N) : MonoidAlgebra.mapDomainRingHom R (f.comp g) = (MonoidAlgebra.mapDomainRingHom R f).comp (MonoidAlgebra.mapDomainRingHom R g)
• AddMonoidAlgebra.mapRingHom_comp_mapDomainRingHom 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} {N : Type u_7} [Semiring R] [Semiring S] [AddMonoid M] [AddMonoid N] (f : R →+* S) (g : M →+ N) : (AddMonoidAlgebra.mapRingHom N f).comp (AddMonoidAlgebra.mapDomainRingHom R g) = (AddMonoidAlgebra.mapDomainRingHom S g).comp (AddMonoidAlgebra.mapRingHom M f)
• MonoidAlgebra.mapRangeRingHom_comp_mapDomainRingHom 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} {N : Type u_7} [Semiring R] [Semiring S] [Monoid M] [Monoid N] (f : R →+* S) (g : M →* N) : (MonoidAlgebra.mapRingHom N f).comp (MonoidAlgebra.mapDomainRingHom R g) = (MonoidAlgebra.mapDomainRingHom S g).comp (MonoidAlgebra.mapRingHom M f)
• MonoidAlgebra.mapRingHom_comp_mapDomainRingHom 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} {N : Type u_7} [Semiring R] [Semiring S] [Monoid M] [Monoid N] (f : R →+* S) (g : M →* N) : (MonoidAlgebra.mapRingHom N f).comp (MonoidAlgebra.mapDomainRingHom R g) = (MonoidAlgebra.mapDomainRingHom S g).comp (MonoidAlgebra.mapRingHom M f)
• Ideal.map_map 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_3} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : Ideal.map g (Ideal.map f I) = Ideal.map (g.comp f) I
• RingHom.ker_equiv_comp 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : R →+* S) (e : S ≃+* T) : RingHom.ker (e.toRingHom.comp f) = RingHom.ker f
• RingHom.comap_ker 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S →+* R) (g : T →+* S) : Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)
• Ideal.comap_comap 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_3} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : Ideal.comap f (Ideal.comap g I) = Ideal.comap (g.comp f) I
• RingHom.ker_comp_of_injective 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} {T : Type w} [Ring R] [Semiring S] [Semiring T] (g : T →+* R) {f : R →+* S} (hf : Function.Injective ⇑f) : RingHom.ker (f.comp g) = RingHom.ker g
• RingHom.liftOfSurjective_comp 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }) : ((f.liftOfSurjective hf) g).comp f = ↑g
• RingHom.liftOfRightInverse_comp 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : { g // RingHom.ker f ≤ RingHom.ker g }) : ((f.liftOfRightInverse f_inv hf) g).comp f = ↑g
• RingHom.eq_liftOfSurjective 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) : h = (f.liftOfSurjective hf) ⟨g, hg⟩
• RingHom.eq_liftOfRightInverse 📋 Mathlib.RingTheory.Ideal.Maps
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) : h = (f.liftOfRightInverse f_inv hf) ⟨g, hg⟩
• RingHom.mapMatrix_comp 📋 Mathlib.Data.Matrix.Basic
  {m : Type u_2} {α : Type u_8} {β : Type u_9} {γ : Type u_10} [Fintype m] [DecidableEq m] [NonAssocSemiring α] [NonAssocSemiring β] [NonAssocSemiring γ] (f : β →+* γ) (g : α →+* β) : f.mapMatrix.comp g.mapMatrix = (f.comp g).mapMatrix
• Matrix.algebraMap_eq_diagonalRingHom 📋 Mathlib.Data.Matrix.Basic
  {n : Type u_3} {R : Type u_4} {α : Type u_8} [Fintype n] [DecidableEq n] [CommSemiring R] [Semiring α] [Algebra R α] : algebraMap R (Matrix n n α) = (Matrix.diagonalRingHom n α).comp (algebraMap R (n → α))
• Subalgebra.algebraMap_eq 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
  {α : Type u_1} {R : Type u_3} {A : Type u_4} [CommSemiring R] [CommSemiring A] [Semiring α] [Algebra R A] [Algebra A α] (S : Subalgebra R A) : algebraMap (↥S) α = (algebraMap A α).comp ↑S.val
• RingHom.star_def 📋 Mathlib.Algebra.Star.Basic
  {R : Type u} [CommSemiring R] [StarRing R] {S : Type u_1} [NonAssocSemiring S] (f : S →+* R) : star f = (starRingEnd R).comp f
• RingHom.map_fieldRange 📋 Mathlib.Algebra.Field.Subfield.Basic
  {K : Type u} {L : Type v} {M : Type w} [DivisionRing K] [DivisionRing L] [DivisionRing M] (g : L →+* M) (f : K →+* L) : Subfield.map g f.fieldRange = (g.comp f).fieldRange
• Subfield.comap_comap 📋 Mathlib.Algebra.Field.Subfield.Basic
  {K : Type u} {L : Type v} {M : Type w} [DivisionRing K] [DivisionRing L] [DivisionRing M] (s : Subfield M) (g : L →+* M) (f : K →+* L) : Subfield.comap f (Subfield.comap g s) = Subfield.comap (g.comp f) s
• Subfield.map_map 📋 Mathlib.Algebra.Field.Subfield.Basic
  {K : Type u} {L : Type v} {M : Type w} [DivisionRing K] [DivisionRing L] [DivisionRing M] (s : Subfield K) (g : L →+* M) (f : K →+* L) : Subfield.map g (Subfield.map f s) = Subfield.map (g.comp f) s
• IsLocalization.lift_of_comp 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] (j : S →+* P) : IsLocalization.lift ⋯ = j
• IsLocalization.ringHom_ext 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] {P : Type u_4} [Semiring P] ⦃j k : S →+* P⦄ (h : j.comp (algebraMap R S) = k.comp (algebraMap R S)) : j = k
• IsLocalization.isUnit_comp 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] (j : S →+* P) (y : ↥M) : IsUnit ((j.comp (algebraMap R S)) ↑y)
• IsLocalization.lift_comp 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} (hg : ∀ (y : ↥M), IsUnit (g ↑y)) : (IsLocalization.lift hg).comp (algebraMap R S) = g
• IsLocalization.map_comp 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] (hy : M ≤ Submonoid.comap g T) : (IsLocalization.map Q g hy).comp (algebraMap R S) = (algebraMap P Q).comp g
• IsLocalization.isLocalization_of_base_ringEquiv 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] (h : R ≃+* P) : IsLocalization (Submonoid.map h M) S
• IsLocalization.isLocalization_iff_of_base_ringEquiv 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) (S : Type u_2) [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (h : R ≃+* P) : IsLocalization M S ↔ IsLocalization (Submonoid.map h M) S
• IsLocalization.map_comp_map 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] (hy : M ≤ Submonoid.comap g T) {A : Type u_5} [CommSemiring A] {U : Submonoid A} {W : Type u_6} [CommSemiring W] [Algebra A W] [IsLocalization U W] {l : P →+* A} (hl : T ≤ Submonoid.comap l U) : (IsLocalization.map W l hl).comp (IsLocalization.map Q g hy) = IsLocalization.map W (l.comp g) ⋯
• IsLocalization.map_map 📋 Mathlib.RingTheory.Localization.Defs
  {R : Type u_1} [CommSemiring R] {M : Submonoid R} {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [IsLocalization M S] {g : R →+* P} {T : Submonoid P} {Q : Type u_4} [CommSemiring Q] [Algebra P Q] [IsLocalization T Q] (hy : M ≤ Submonoid.comap g T) {A : Type u_5} [CommSemiring A] {U : Submonoid A} {W : Type u_6} [CommSemiring W] [Algebra A W] [IsLocalization U W] {l : P →+* A} (hl : T ≤ Submonoid.comap l U) (x : S) : (IsLocalization.map W l hl) ((IsLocalization.map Q g hy) x) = (IsLocalization.map W (l.comp g) ⋯) x
• IsLocalization.isLocalization_iff_of_ringEquiv 📋 Mathlib.RingTheory.Localization.Basic
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (h : S ≃+* P) : IsLocalization M S ↔ IsLocalization M P
• IsLocalization.bijective 📋 Mathlib.RingTheory.Localization.Basic
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] [IsLocalization M S] {Q : Type u_4} [CommSemiring Q] [Algebra R Q] [IsLocalization M Q] (f : S →+* Q) (hf : f.comp (algebraMap R S) = algebraMap R Q) : Function.Bijective ⇑f
• IsLocalization.algEquiv_comp_algebraMap 📋 Mathlib.RingTheory.Localization.Basic
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] (Rₘ : Type u_4) (Sₙ : Type u_5) (Rₘ' : Type u_6) (Sₙ' : Type u_7) [CommSemiring Rₘ] [CommSemiring Sₙ] [CommSemiring Rₘ'] [CommSemiring Sₙ'] [Algebra R Rₘ] [Algebra S Sₙ] [Algebra R Rₘ'] [Algebra S Sₙ'] [Algebra R Sₙ] [Algebra Rₘ Sₙ] [Algebra Rₘ' Sₙ'] [Algebra R Sₙ'] (N : Submonoid S) [IsLocalization M Rₘ] [IsLocalization N Sₙ] [IsLocalization M Rₘ'] [IsLocalization N Sₙ'] [IsScalarTower R Rₘ Sₙ] [IsScalarTower R S Sₙ] [IsScalarTower R Rₘ' Sₙ'] [IsScalarTower R S Sₙ'] : (↑(IsLocalization.algEquiv N Sₙ Sₙ')).comp (algebraMap Rₘ Sₙ) = (algebraMap Rₘ' Sₙ').comp ↑(IsLocalization.algEquiv M Rₘ Rₘ')
• IsLocalization.algEquiv_comp_algebraMap_apply 📋 Mathlib.RingTheory.Localization.Basic
  {R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] {Rₘ : Type u_4} (Sₙ : Type u_5) (Rₘ' : Type u_6) (Sₙ' : Type u_7) [CommSemiring Rₘ] [CommSemiring Sₙ] [CommSemiring Rₘ'] [CommSemiring Sₙ'] [Algebra R Rₘ] [Algebra S Sₙ] [Algebra R Rₘ'] [Algebra S Sₙ'] [Algebra R Sₙ] [Algebra Rₘ Sₙ] [Algebra Rₘ' Sₙ'] [Algebra R Sₙ'] (N : Submonoid S) [IsLocalization M Rₘ] [IsLocalization N Sₙ] [IsLocalization M Rₘ'] [IsLocalization N Sₙ'] [IsScalarTower R Rₘ Sₙ] [IsScalarTower R S Sₙ] [IsScalarTower R Rₘ' Sₙ'] [IsScalarTower R S Sₙ'] (x : Rₘ) : ((↑(IsLocalization.algEquiv N Sₙ Sₙ')).comp (algebraMap Rₘ Sₙ)) x = ((algebraMap Rₘ' Sₙ').comp ↑(IsLocalization.algEquiv M Rₘ Rₘ')) x
• IsFractionRing.injective_comp_algebraMap 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] {L : Type u_7} [Field L] [Algebra A K] [IsFractionRing A K] : Function.Injective fun f => f.comp (algebraMap A K)
• IsFractionRing.isFractionRing_iff_of_base_ringEquiv 📋 Mathlib.RingTheory.Localization.FractionRing
  {R : Type u_1} [CommRing R] (S : Type u_2) [CommRing S] [Algebra R S] {P : Type u_3} [CommRing P] (h : R ≃+* P) : IsFractionRing R S ↔ IsFractionRing P S
• IsFractionRing.ringHom_fieldRange_eq_of_comp_eq 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_8} [Field L] {g : A →+* L} {f : K →+* L} (h : f.comp (algebraMap A K) = g) : f.fieldRange = Subfield.closure ↑g.range
• IsFractionRing.ringHom_fieldRange_eq_of_comp_eq_of_range_eq 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_8} [Field L] {g : A →+* L} {f : K →+* L} (h : f.comp (algebraMap A K) = g) {s : Set L} (hs : g.range = Subring.closure s) : f.fieldRange = Subfield.closure s
• IsFractionRing.semilinearEquivOfRingEquiv_comp 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_8} {B : Type u_9} (K : Type u_10) (L : Type u_11) [CommRing A] [CommRing B] [CommRing K] [CommRing L] [Algebra A K] [IsFractionRing A K] [Algebra B L] [IsFractionRing B L] (f : A ≃+* B) {C : Type u_12} (M : Type u_13) [CommRing C] [CommRing M] [Algebra C M] [IsFractionRing C M] (g : B ≃+* C) : have this := ⋯; have this_1 := ⋯; IsFractionRing.semilinearEquivOfRingEquiv K M (f.trans g) = (IsFractionRing.semilinearEquivOfRingEquiv K L f).trans (IsFractionRing.semilinearEquivOfRingEquiv L M g)
• Algebra.TensorProduct.algebraMap_def 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] : algebraMap S (TensorProduct R A B) = Algebra.TensorProduct.includeLeftRingHom.comp (algebraMap S A)
• Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] : Algebra.TensorProduct.includeLeftRingHom.comp (algebraMap R A) = Algebra.TensorProduct.includeRight.comp (algebraMap R B)
• Algebra.TensorProduct.ringHom_ext 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {C : Type u_2} [Semiring C] {f g : TensorProduct R A B →+* C} (h₁ : f.comp Algebra.TensorProduct.includeLeftRingHom = g.comp Algebra.TensorProduct.includeLeftRingHom) (h₂ : f.comp Algebra.TensorProduct.includeRight.toRingHom = g.comp Algebra.TensorProduct.includeRight.toRingHom) : f = g
• Algebra.TensorProduct.ringHom_ext_iff 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] {C : Type u_2} [Semiring C] {f g : TensorProduct R A B →+* C} : f = g ↔ f.comp Algebra.TensorProduct.includeLeftRingHom = g.comp Algebra.TensorProduct.includeLeftRingHom ∧ f.comp Algebra.TensorProduct.includeRight.toRingHom = g.comp Algebra.TensorProduct.includeRight.toRingHom
• Algebra.TensorProduct.mapRingHom 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {R' : Type u_6} {S' : Type u_7} {T' : Type u_8} [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra R S] [Algebra R T] [CommSemiring R'] [CommSemiring S'] [CommSemiring T'] [Algebra R' S'] [Algebra R' T'] (fR : R →+* R') (fS : S →+* S') (fT : T →+* T') (HS : fS.comp (algebraMap R S) = (algebraMap R' S').comp fR) (HT : fT.comp (algebraMap R T) = (algebraMap R' T').comp fR) : TensorProduct R S T →+* TensorProduct R' S' T'
• Algebra.TensorProduct.mapRingHom_comp_includeLeftRingHom 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {R' : Type u_6} {S' : Type u_7} {T' : Type u_8} [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra R S] [Algebra R T] [CommSemiring R'] [CommSemiring S'] [CommSemiring T'] [Algebra R' S'] [Algebra R' T'] (fR : R →+* R') (fS : S →+* S') (fT : T →+* T') (HS : fS.comp (algebraMap R S) = (algebraMap R' S').comp fR) (HT : fT.comp (algebraMap R T) = (algebraMap R' T').comp fR) : (Algebra.TensorProduct.mapRingHom fR fS fT HS HT).comp Algebra.TensorProduct.includeLeftRingHom = Algebra.TensorProduct.includeLeftRingHom.comp fS
• Algebra.TensorProduct.mapRingHom_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {R' : Type u_6} {S' : Type u_7} {T' : Type u_8} [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra R S] [Algebra R T] [CommSemiring R'] [CommSemiring S'] [CommSemiring T'] [Algebra R' S'] [Algebra R' T'] (fR : R →+* R') (fS : S →+* S') (fT : T →+* T') (HS : fS.comp (algebraMap R S) = (algebraMap R' S').comp fR) (HT : fT.comp (algebraMap R T) = (algebraMap R' T').comp fR) (s : S) (t : T) : (Algebra.TensorProduct.mapRingHom fR fS fT HS HT) (s ⊗ₜ[R] t) = fS s ⊗ₜ[R'] fT t
• Algebra.TensorProduct.rid_comp_includeLeftRingHom 📋 Mathlib.RingTheory.TensorProduct.Maps
  (R : Type uR) (S : Type uS) (A : Type uA) [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] : (↑(Algebra.TensorProduct.rid R S A)).comp Algebra.TensorProduct.includeLeftRingHom = RingHom.id A
• Algebra.TensorProduct.map_comp_includeLeftRingHom 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} {D : Type uD} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [Algebra S C] [IsScalarTower R S C] [Semiring D] [Algebra R D] (f : A →ₐ[S] C) (g : B →ₐ[R] D) : (↑(Algebra.TensorProduct.map f g)).comp Algebra.TensorProduct.includeLeftRingHom = Algebra.TensorProduct.includeLeftRingHom.comp ↑f
• Algebra.TensorProduct.mapRingHom_comp_includeRight 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type u_3} {S : Type u_4} {T : Type u_5} {R' : Type u_6} {S' : Type u_7} {T' : Type u_8} [CommSemiring R] [CommSemiring S] [CommSemiring T] [Algebra R S] [Algebra R T] [CommSemiring R'] [CommSemiring S'] [CommSemiring T'] [Algebra R' S'] [Algebra R' T'] (fR : R →+* R') (fS : S →+* S') (fT : T →+* T') (HS : fS.comp (algebraMap R S) = (algebraMap R' S').comp fR) (HT : fT.comp (algebraMap R T) = (algebraMap R' T').comp fR) : (Algebra.TensorProduct.mapRingHom fR fS fT HS HT).comp ↑Algebra.TensorProduct.includeRight = (↑Algebra.TensorProduct.includeRight).comp fT
• Unitization.algebraMap_eq_inlRingHom_comp 📋 Mathlib.Algebra.Algebra.Unitization
  (S : Type u_1) (R : Type u_2) (A : Type u_3) [CommSemiring S] [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [Algebra S R] [DistribMulAction S A] [IsScalarTower S R A] : algebraMap S (Unitization R A) = (Unitization.inlRingHom R A).comp (algebraMap S R)
• LinearMap.compPMap_apply 📋 Mathlib.LinearAlgebra.LinearPMap
  {R : Type u_1} {S : Type u_2} {T : Type u_3} [Ring R] [Ring S] [Ring T] {σ : R →+* S} {τ : S →+* T} {E : Type u_4} [AddCommGroup E] [Module R E] {F : Type u_5} [AddCommGroup F] [Module S F] {G : Type u_6} [AddCommGroup G] [Module T G] (g : F →ₛₗ[τ] G) (f : E →ₛₗ.[σ] F) (x : ↥(g.compPMap f).domain) : ↑(g.compPMap f) x = g (↑f x)
• AddMonoidAlgebra.mapDomainRingHom_comp_algebraMap 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} {N : Type u_8} [CommSemiring R] [Semiring A] [Algebra R A] [AddMonoid M] [AddMonoid N] (f : M →+ N) : (AddMonoidAlgebra.mapDomainRingHom A f).comp (algebraMap R (AddMonoidAlgebra A M)) = algebraMap R (AddMonoidAlgebra A N)
• MonoidAlgebra.mapDomainRingHom_comp_algebraMap 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} {N : Type u_8} [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [Monoid N] (f : M →* N) : (MonoidAlgebra.mapDomainRingHom A f).comp (algebraMap R (MonoidAlgebra A M)) = algebraMap R (MonoidAlgebra A N)
• AddMonoidAlgebra.mapRingHom_comp_algebraMap 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {S : Type u_2} {M : Type u_7} [CommSemiring R] [CommSemiring S] [AddMonoid M] (f : R →+* S) : (AddMonoidAlgebra.mapRingHom M f).comp (algebraMap R (AddMonoidAlgebra R M)) = (algebraMap S (AddMonoidAlgebra S M)).comp f
• MonoidAlgebra.mapRangeRingHom_comp_algebraMap 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {S : Type u_2} {M : Type u_7} [CommSemiring R] [CommSemiring S] [Monoid M] (f : R →+* S) : (MonoidAlgebra.mapRingHom M f).comp (algebraMap R (MonoidAlgebra R M)) = (algebraMap S (MonoidAlgebra S M)).comp f
• MonoidAlgebra.mapRingHom_comp_algebraMap 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {S : Type u_2} {M : Type u_7} [CommSemiring R] [CommSemiring S] [Monoid M] (f : R →+* S) : (MonoidAlgebra.mapRingHom M f).comp (algebraMap R (MonoidAlgebra R M)) = (algebraMap S (MonoidAlgebra S M)).comp f
• Polynomial.eval₂RingHom_comp_C 📋 Mathlib.Algebra.Polynomial.Eval.Defs
  {R : Type u} {S : Type v} [Semiring R] [CommSemiring S] (f : R →+* S) (x : S) : (Polynomial.eval₂RingHom f x).comp Polynomial.C = f
• Polynomial.mapRingHom_comp_C 📋 Mathlib.Algebra.Polynomial.Eval.Defs
  {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) : (Polynomial.mapRingHom f).comp Polynomial.C = Polynomial.C.comp f
• Polynomial.map_map 📋 Mathlib.Algebra.Polynomial.Eval.Coeff
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] (f : R →+* S) [Semiring T] (g : S →+* T) (p : Polynomial R) : Polynomial.map g (Polynomial.map f p) = Polynomial.map (g.comp f) p
• Polynomial.eval₂_map 📋 Mathlib.Algebra.Polynomial.Eval.Coeff
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] {p : Polynomial R} [Semiring S] (f : R →+* S) [Semiring T] (g : S →+* T) (x : T) : Polynomial.eval₂ g x (Polynomial.map f p) = Polynomial.eval₂ (g.comp f) x p
• Polynomial.hom_eval₂ 📋 Mathlib.Algebra.Polynomial.Eval.Coeff
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] (p : Polynomial R) [Semiring S] [Semiring T] (f : R →+* S) (g : S →+* T) (x : S) : g (Polynomial.eval₂ f x p) = Polynomial.eval₂ (g.comp f) (g x) p
• Polynomial.mapRingHom_comp 📋 Mathlib.Algebra.Polynomial.Eval.Coeff
  {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S →+* T) (g : R →+* S) : (Polynomial.mapRingHom f).comp (Polynomial.mapRingHom g) = Polynomial.mapRingHom (f.comp g)
• Polynomial.ringHom_ext' 📋 Mathlib.Algebra.Polynomial.Monomial
  {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] {f g : Polynomial R →+* S} (h₁ : f.comp Polynomial.C = g.comp Polynomial.C) (h₂ : f Polynomial.X = g Polynomial.X) : f = g
• Polynomial.ringHom_ext'_iff 📋 Mathlib.Algebra.Polynomial.Monomial
  {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] {f g : Polynomial R →+* S} : f = g ↔ f.comp Polynomial.C = g.comp Polynomial.C ∧ f Polynomial.X = g Polynomial.X
• Polynomial.aeval_eq_aeval_map 📋 Mathlib.Algebra.Polynomial.AlgebraMap
  {R : Type u} {S : Type v} {T : Type w} [CommSemiring R] [Semiring S] [CommSemiring T] [Algebra R S] [Algebra T S] {φ : R →+* T} (h : (algebraMap T S).comp φ = algebraMap R S) (p : Polynomial R) (a : S) : (Polynomial.aeval a) p = (Polynomial.aeval a) (Polynomial.map φ p)
• Polynomial.map_aeval_eq_aeval_map 📋 Mathlib.Algebra.Polynomial.AlgebraMap
  {R : Type u} [CommSemiring R] {S : Type u_3} {T : Type u_4} {U : Type u_5} [Semiring S] [CommSemiring T] [Semiring U] [Algebra R S] [Algebra T U] {φ : R →+* T} {ψ : S →+* U} (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) (p : Polynomial R) (a : S) : ψ ((Polynomial.aeval a) p) = (Polynomial.aeval (ψ a)) (Polynomial.map φ p)
• Polynomial.aevalTower_comp_C 📋 Mathlib.Algebra.Polynomial.AlgebraMap
  {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R →ₐ[S] A') (y : A') : (↑(Polynomial.aevalTower g y)).comp Polynomial.C = ↑g
• Polynomial.aevalTower_comp_algebraMap 📋 Mathlib.Algebra.Polynomial.AlgebraMap
  {R : Type u} {S : Type v} {A' : Type u_1} [CommSemiring R] [CommSemiring A'] [CommSemiring S] [Algebra S R] [Algebra S A'] (g : R →ₐ[S] A') (y : A') : (↑(Polynomial.aevalTower g y)).comp (algebraMap R (Polynomial R)) = ↑g
• MvPolynomial.constantCoeff_comp_C 📋 Mathlib.Algebra.MvPolynomial.Basic
  (R : Type u) (σ : Type u_1) [CommSemiring R] : MvPolynomial.constantCoeff.comp MvPolynomial.C = RingHom.id R
• MvPolynomial.constantCoeff_comp_algebraMap 📋 Mathlib.Algebra.MvPolynomial.Basic
  (R : Type u) (σ : Type u_1) [CommSemiring R] : MvPolynomial.constantCoeff.comp (algebraMap R (MvPolynomial σ R)) = RingHom.id R
• MvPolynomial.ringHom_ext' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {f g : MvPolynomial σ R →+* A} (hC : f.comp MvPolynomial.C = g.comp MvPolynomial.C) (hX : ∀ (i : σ), f (MvPolynomial.X i) = g (MvPolynomial.X i)) : f = g
• MvPolynomial.ringHom_ext'_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] {f g : MvPolynomial σ R →+* A} : f = g ↔ f.comp MvPolynomial.C = g.comp MvPolynomial.C ∧ ∀ (i : σ), f (MvPolynomial.X i) = g (MvPolynomial.X i)
• MvPolynomial.is_id 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (f : MvPolynomial σ R →+* MvPolynomial σ R) (hC : f.comp MvPolynomial.C = MvPolynomial.C) (hX : ∀ (n : σ), f (MvPolynomial.X n) = MvPolynomial.X n) (p : MvPolynomial σ R) : f p = p
• MvPolynomial.hom_eq_hom 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [Semiring S₂] (f g : MvPolynomial σ R →+* S₂) (hC : f.comp MvPolynomial.C = g.comp MvPolynomial.C) (hX : ∀ (n : σ), f (MvPolynomial.X n) = g (MvPolynomial.X n)) (p : MvPolynomial σ R) : f p = g p
• MvPolynomial.eval₂Hom_zero' 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₂] (f : R →+* S₂) : (MvPolynomial.eval₂Hom f fun x => 0) = f.comp MvPolynomial.constantCoeff
• MvPolynomial.eval₂Hom_zero 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₂] (f : R →+* S₂) : MvPolynomial.eval₂Hom f 0 = f.comp MvPolynomial.constantCoeff
• MvPolynomial.map_eq_eval₂Hom_C_comp 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : MvPolynomial.map f = MvPolynomial.eval₂Hom (MvPolynomial.C.comp f) MvPolynomial.X
• MvPolynomial.hom_eval₂ 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (p : MvPolynomial σ R) (f : R →+* S₁) (φ : S₁ →+* S₂) (g : σ → S₁) : φ (MvPolynomial.eval₂ f g p) = MvPolynomial.eval₂ (φ.comp f) (fun i => φ (g i)) p
• MvPolynomial.constantCoeff_comp_map 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : MvPolynomial.constantCoeff.comp (MvPolynomial.map f) = f.comp MvPolynomial.constantCoeff
• MvPolynomial.eval₂_comp_left 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {S₂ : Type u_2} [CommSemiring S₂] (k : S₁ →+* S₂) (f : R →+* S₁) (g : σ → S₁) (p : MvPolynomial σ R) : k (MvPolynomial.eval₂ f g p) = MvPolynomial.eval₂ (k.comp f) (⇑k ∘ g) p
• MvPolynomial.comp_eval₂Hom 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₁) (φ : S₁ →+* S₂) : φ.comp (MvPolynomial.eval₂Hom f g) = MvPolynomial.eval₂Hom (φ.comp f) fun i => φ (g i)
• MvPolynomial.eval₂_map 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₂) (φ : S₁ →+* S₂) (p : MvPolynomial σ R) : MvPolynomial.eval₂ φ g ((MvPolynomial.map f) p) = MvPolynomial.eval₂ (φ.comp f) g p
• MvPolynomial.eval₂_map_comp_C 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {ι : Type u_2} (f : R →+* S₁) (h : ι → MvPolynomial σ S₁) (p : MvPolynomial ι R) : MvPolynomial.eval₂ ((MvPolynomial.map f).comp MvPolynomial.C) h p = MvPolynomial.eval₂ MvPolynomial.C h ((MvPolynomial.map f) p)
• MvPolynomial.eval_eval₂ 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {σ : Type u_1} [CommSemiring R] {S : Type u_2} {τ : Type u_3} {x : τ → S} [CommSemiring S] (f : R →+* MvPolynomial τ S) (g : σ → MvPolynomial τ S) (p : MvPolynomial σ R) : (MvPolynomial.eval x) (MvPolynomial.eval₂ f g p) = MvPolynomial.eval₂ ((MvPolynomial.eval x).comp f) (fun s => (MvPolynomial.eval x) (g s)) p
• MvPolynomial.map_eval₂Hom 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₁) (φ : S₁ →+* S₂) (p : MvPolynomial σ R) : φ ((MvPolynomial.eval₂Hom f g) p) = (MvPolynomial.eval₂Hom (φ.comp f) fun i => φ (g i)) p
• MvPolynomial.map_aeval 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [Algebra R S₁] {B : Type u_2} [CommSemiring B] (g : σ → S₁) (φ : S₁ →+* B) (p : MvPolynomial σ R) : φ ((MvPolynomial.aeval g) p) = (MvPolynomial.eval₂Hom (φ.comp (algebraMap R S₁)) fun i => φ (g i)) p
• MvPolynomial.eval₂Hom_map_hom 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₂) (φ : S₁ →+* S₂) (p : MvPolynomial σ R) : (MvPolynomial.eval₂Hom φ g) ((MvPolynomial.map f) p) = (MvPolynomial.eval₂Hom (φ.comp f) g) p
• MvPolynomial.map_map 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) [CommSemiring S₂] (g : S₁ →+* S₂) (p : MvPolynomial σ R) : (MvPolynomial.map g) ((MvPolynomial.map f) p) = (MvPolynomial.map (g.comp f)) p
• MvPolynomial.aevalTower_comp_C 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {σ : Type u_1} [CommSemiring R] {S : Type u_2} {A : Type u_3} [CommSemiring S] [CommSemiring A] [Algebra S R] [Algebra S A] (g : R →ₐ[S] A) (y : σ → A) : (↑(MvPolynomial.aevalTower g y)).comp MvPolynomial.C = ↑g
• MvPolynomial.aevalTower_comp_algebraMap 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {σ : Type u_1} [CommSemiring R] {S : Type u_2} {A : Type u_3} [CommSemiring S] [CommSemiring A] [Algebra S R] [Algebra S A] (g : R →ₐ[S] A) (y : σ → A) : (↑(MvPolynomial.aevalTower g y)).comp (algebraMap R (MvPolynomial σ R)) = ↑g
• MvPolynomial.map_comp_rename 📋 Mathlib.Algebra.MvPolynomial.Rename
  {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → τ) : (MvPolynomial.map f).comp (MvPolynomial.rename g).toRingHom = (MvPolynomial.rename g).comp (MvPolynomial.map f)
• MvPolynomial.eval_comp_toMvPolynomial 📋 Mathlib.Algebra.MvPolynomial.Equiv
  {R : Type u_1} {σ : Type u_3} [CommSemiring R] (f : σ → R) (i : σ) : (MvPolynomial.eval f).comp ↑(Polynomial.toMvPolynomial i) = Polynomial.evalRingHom (f i)
• MvPolynomial.mvPolynomialEquivMvPolynomial 📋 Mathlib.Algebra.MvPolynomial.Equiv
  (R : Type u) (S₁ : Type v) (S₂ : Type w) (S₃ : Type x) [CommSemiring R] [CommSemiring S₃] (f : MvPolynomial S₁ R →+* MvPolynomial S₂ S₃) (g : MvPolynomial S₂ S₃ →+* MvPolynomial S₁ R) (hfgC : (f.comp g).comp MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), f (g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : (g.comp f).comp MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), g (f (MvPolynomial.X n)) = MvPolynomial.X n) : MvPolynomial S₁ R ≃+* MvPolynomial S₂ S₃
• MvPolynomial.mvPolynomialEquivMvPolynomial_apply 📋 Mathlib.Algebra.MvPolynomial.Equiv
  (R : Type u) (S₁ : Type v) (S₂ : Type w) (S₃ : Type x) [CommSemiring R] [CommSemiring S₃] (f : MvPolynomial S₁ R →+* MvPolynomial S₂ S₃) (g : MvPolynomial S₂ S₃ →+* MvPolynomial S₁ R) (hfgC : (f.comp g).comp MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), f (g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : (g.comp f).comp MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), g (f (MvPolynomial.X n)) = MvPolynomial.X n) (a : MvPolynomial S₁ R) : (MvPolynomial.mvPolynomialEquivMvPolynomial R S₁ S₂ S₃ f g hfgC hfgX hgfC hgfX) a = f a
• MvPolynomial.mvPolynomialEquivMvPolynomial_symm_apply 📋 Mathlib.Algebra.MvPolynomial.Equiv
  (R : Type u) (S₁ : Type v) (S₂ : Type w) (S₃ : Type x) [CommSemiring R] [CommSemiring S₃] (f : MvPolynomial S₁ R →+* MvPolynomial S₂ S₃) (g : MvPolynomial S₂ S₃ →+* MvPolynomial S₁ R) (hfgC : (f.comp g).comp MvPolynomial.C = MvPolynomial.C) (hfgX : ∀ (n : S₂), f (g (MvPolynomial.X n)) = MvPolynomial.X n) (hgfC : (g.comp f).comp MvPolynomial.C = MvPolynomial.C) (hgfX : ∀ (n : S₁), g (f (MvPolynomial.X n)) = MvPolynomial.X n) (a : MvPolynomial S₂ S₃) : (MvPolynomial.mvPolynomialEquivMvPolynomial R S₁ S₂ S₃ f g hfgC hfgX hgfC hgfX).symm a = g a
• MvPolynomial.finSuccEquiv_apply 📋 Mathlib.Algebra.MvPolynomial.Equiv
  (R : Type u) [CommSemiring R] (n : ℕ) (p : MvPolynomial (Fin (n + 1)) R) : (MvPolynomial.finSuccEquiv R n) p = (MvPolynomial.eval₂Hom (Polynomial.C.comp MvPolynomial.C) fun i => Fin.cases Polynomial.X (fun k => Polynomial.C (MvPolynomial.X k)) i) p
• MvPolynomial.finSuccEquiv_comp_C_eq_C 📋 Mathlib.Algebra.MvPolynomial.Equiv
  {R : Type u} [CommSemiring R] (n : ℕ) : (↑(MvPolynomial.finSuccEquiv R n).symm).comp (Polynomial.C.comp MvPolynomial.C) = MvPolynomial.C
• MvPolynomial.finSuccEquiv_eq 📋 Mathlib.Algebra.MvPolynomial.Equiv
  (R : Type u) [CommSemiring R] (n : ℕ) : ↑(MvPolynomial.finSuccEquiv R n) = MvPolynomial.eval₂Hom (Polynomial.C.comp MvPolynomial.C) fun i => Fin.cases Polynomial.X (fun k => Polynomial.C (MvPolynomial.X k)) i
• MvPolynomial.map_mvPolynomial_eq_eval₂ 📋 Mathlib.RingTheory.Polynomial.Basic
  {R : Type u} {σ : Type v} [CommRing R] {S : Type u_2} [CommSemiring S] [Finite σ] (ϕ : MvPolynomial σ R →+* S) (p : MvPolynomial σ R) : ϕ p = MvPolynomial.eval₂ (ϕ.comp MvPolynomial.C) (fun s => ϕ (MvPolynomial.X s)) p
• Polynomial.eval₂_restriction 📋 Mathlib.RingTheory.Polynomial.Basic
  {R : Type u} {S : Type u_1} [Ring R] [Semiring S] {f : R →+* S} {x : S} {p : Polynomial R} : Polynomial.eval₂ f x p = Polynomial.eval₂ (f.comp (Subring.closure ↑p.coeffs).subtype) x p.restriction
• Ideal.Quotient.mk_comp_algebraMap 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] : (Ideal.Quotient.mk I).comp (algebraMap R₁ A) = algebraMap R₁ (A ⧸ I)
• Ideal.Quotient.alg_map_eq 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  (R₁ : Type u_1) [CommSemiring R₁] {A : Type u_5} [CommRing A] [Algebra R₁ A] (I : Ideal A) : algebraMap R₁ (A ⧸ I) = (algebraMap A (A ⧸ I)).comp (algebraMap R₁ A)
• Ideal.quotientMap_comp_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {S : Type v} [Ring S] {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : R →+* S} (H : J ≤ Ideal.comap f I) : (Ideal.quotientMap I f H).comp (Ideal.Quotient.mk J) = (Ideal.Quotient.mk I).comp f
• RingHom.quotientKerEquivOfSurjective_symm_comp 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) : (RingHom.quotientKerEquivOfSurjective hf).symm.toRingHom.comp f = Ideal.Quotient.mk (RingHom.ker f)
• Ideal.fst_comp_quotientInfEquivQuotientProd 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) : (RingHom.fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientInfEquivQuotientProd J coprime) = Ideal.Quotient.factor ⋯
• Ideal.snd_comp_quotientInfEquivQuotientProd 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) : (RingHom.snd (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientInfEquivQuotientProd J coprime) = Ideal.Quotient.factor ⋯
• Ideal.comp_quotientMap_eq_of_comp_eq 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {S : Type v} [Ring S] {R' : Type u_5} {S' : Type u_6} [Ring R'] [Ring S'] {f : R →+* S} {f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f) (I : Ideal S') [I.IsTwoSided] : have leq := ⋯; (Ideal.quotientMap I g' ⋯).comp (Ideal.quotientMap (Ideal.comap g' I) f ⋯) = (Ideal.quotientMap I f' ⋯).comp (Ideal.quotientMap (Ideal.comap f' I) g leq)
• Ideal.snd_comp_quotientMulEquivQuotientProd 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) : (RingHom.snd (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime) = Ideal.Quotient.factor ⋯
• Ideal.fst_comp_quotientMulEquivQuotientProd 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) : (RingHom.fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime) = Ideal.Quotient.factor ⋯
• DoubleQuot.quotQuotEquivQuotOfLE_comp_quotQuotMk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [CommRing R] {I J : Ideal R} (h : I ≤ J) : (↑(DoubleQuot.quotQuotEquivQuotOfLE h)).comp (DoubleQuot.quotQuotMk I J) = Ideal.Quotient.mk J
• DoubleQuot.quotQuotEquivQuotOfLE_symm_comp_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [CommRing R] {I J : Ideal R} (h : I ≤ J) : (↑(DoubleQuot.quotQuotEquivQuotOfLE h).symm).comp (Ideal.Quotient.mk J) = DoubleQuot.quotQuotMk I J
• DoubleQuot.quotQuotEquivComm_comp_quotQuotMk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [CommRing R] (I J : Ideal R) : (↑(DoubleQuot.quotQuotEquivComm I J)).comp (DoubleQuot.quotQuotMk I J) = DoubleQuot.quotQuotMk J I
• RingHom.FiniteType.of_comp_finiteType 📋 Mathlib.RingTheory.FiniteType
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {f : A →+* B} {g : B →+* C} (h : (g.comp f).FiniteType) : g.FiniteType
• RingHom.FiniteType.comp 📋 Mathlib.RingTheory.FiniteType
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {g : B →+* C} {f : A →+* B} (hg : g.FiniteType) (hf : f.FiniteType) : (g.comp f).FiniteType
• RingHom.FiniteType.comp_surjective 📋 Mathlib.RingTheory.FiniteType
  {A : Type u_1} {B : Type u_2} {C : Type u_3} [CommRing A] [CommRing B] [CommRing C] {f : A →+* B} {g : B →+* C} (hf : f.FiniteType) (hg : Function.Surjective ⇑g) : (g.comp f).FiniteType
• ZMod.castHom_comp 📋 Mathlib.Data.ZMod.Basic
  {n m d : ℕ} (hm : n ∣ m) (hd : m ∣ d) : (ZMod.castHom hm (ZMod n)).comp (ZMod.castHom hd (ZMod m)) = ZMod.castHom ⋯ (ZMod n)
• iterateFrobenius_add 📋 Mathlib.Algebra.CharP.Frobenius
  (R : Type u_1) [CommSemiring R] (p m n : ℕ) [ExpChar R p] : iterateFrobenius R p (m + n) = (iterateFrobenius R p m).comp (iterateFrobenius R p n)
• RingHom.frobenius_comm 📋 Mathlib.Algebra.CharP.Frobenius
  {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] (g : R →+* S) (p : ℕ) [ExpChar R p] [ExpChar S p] : g.comp (frobenius R p) = (frobenius S p).comp g
• RingHom.iterateFrobenius_comm 📋 Mathlib.Algebra.CharP.Frobenius
  {R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] (g : R →+* S) (p : ℕ) [ExpChar R p] [ExpChar S p] (n : ℕ) : g.comp (iterateFrobenius R p n) = (iterateFrobenius S p n).comp g
• IsAlgebraic.ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) {a : A} (halg : IsAlgebraic R a) (hf : Function.Injective ⇑f) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : IsAlgebraic S (g a)
• Transcendental.of_ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) {a : A} (H : Transcendental S (g a)) (hf : Function.Injective ⇑f) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : Transcendental R a
• Algebra.IsAlgebraic.of_ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) [Algebra.IsAlgebraic S B] (hf : Function.Surjective ⇑f) (hg : Function.Injective ⇑g) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : Algebra.IsAlgebraic R A
• Algebra.IsAlgebraic.ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) [Algebra.IsAlgebraic R A] (hf : Function.Injective ⇑f) (hg : Function.Surjective ⇑g) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : Algebra.IsAlgebraic S B
• Algebra.Transcendental.of_ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) [H : Algebra.Transcendental S B] (hf : Function.Injective ⇑f) (hg : Function.Surjective ⇑g) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : Algebra.Transcendental R A
• Algebra.Transcendental.ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) [H : Algebra.Transcendental R A] (hf : Function.Surjective ⇑f) (hg : Function.Injective ⇑g) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : Algebra.Transcendental S B
• IsAlgebraic.of_ringHom_of_comp_eq 📋 Mathlib.RingTheory.Algebraic.Basic
  {R : Type u} {S : Type u_1} {A : Type v} [CommRing R] [CommRing S] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra S B] {FRS : Type u_3} {FAB : Type u_4} [FunLike FRS R S] [RingHomClass FRS R S] [FunLike FAB A B] [RingHomClass FAB A B] (f : FRS) (g : FAB) {a : A} (halg : IsAlgebraic S (g a)) (hf : Function.Surjective ⇑f) (hg : Function.Injective ⇑g) (h : (algebraMap S B).comp ↑f = (↑g).comp (algebraMap R A)) : IsAlgebraic R a

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 ce5dd8c