statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
refl_to_alg_hom : ↑(refl : A₁ ≃ₐ[R] A₁) = alg_hom.id R A₁ | rfl | lemma | alg_equiv.refl_to_alg_hom | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_refl : ⇑(refl : A₁ ≃ₐ[R] A₁) = id | rfl | lemma | alg_equiv.coe_refl | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : A₁ ≃ₐ[R] A₂) : A₂ ≃ₐ[R] A₁ | { commutes' := λ r, by { rw ←e.to_ring_equiv.symm_apply_apply (algebra_map R A₁ r), congr,
change _ = e _, rw e.commutes, },
..e.to_ring_equiv.symm, } | def | alg_equiv.symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"algebra_map"
] | Algebra equivalences are symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (e : A₁ ≃ₐ[R] A₂) : A₂ → A₁ | e.symm
initialize_simps_projections alg_equiv (to_fun → apply, inv_fun → symm_apply) | def | alg_equiv.simps.symm_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv",
"inv_fun"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_apply_coe_coe_symm_apply {F : Type*} [alg_equiv_class F R A₁ A₂]
(f : F) (x : A₂) : f ((f : A₁ ≃ₐ[R] A₂).symm x) = x | equiv_like.right_inv f x | lemma | alg_equiv.coe_apply_coe_coe_symm_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_symm_apply_coe_apply {F : Type*} [alg_equiv_class F R A₁ A₂]
(f : F) (x : A₁) : (f : A₁ ≃ₐ[R] A₂).symm (f x) = x | equiv_like.left_inv f x | lemma | alg_equiv.coe_coe_symm_apply_coe_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_eq_symm {e : A₁ ≃ₐ[R] A₂} : e.inv_fun = e.symm | rfl | lemma | alg_equiv.inv_fun_eq_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (e : A₁ ≃ₐ[R] A₂) : e.symm.symm = e | by { ext, refl, } | lemma | alg_equiv.symm_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_bijective : function.bijective (symm : (A₁ ≃ₐ[R] A₂) → (A₂ ≃ₐ[R] A₁)) | equiv.bijective ⟨symm, symm, symm_symm, symm_symm⟩ | lemma | alg_equiv.symm_bijective | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"equiv.bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe' (e : A₁ ≃ₐ[R] A₂) (f h₁ h₂ h₃ h₄ h₅) :
(⟨f, e, h₁, h₂, h₃, h₄, h₅⟩ : A₂ ≃ₐ[R] A₁) = e.symm | symm_bijective.injective $ ext $ λ x, rfl | lemma | alg_equiv.mk_coe' | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_mk (f f') (h₁ h₂ h₃ h₄ h₅) :
(⟨f, f', h₁, h₂, h₃, h₄, h₅⟩ : A₁ ≃ₐ[R] A₂).symm =
{ to_fun := f', inv_fun := f,
..(⟨f, f', h₁, h₂, h₃, h₄, h₅⟩ : A₁ ≃ₐ[R] A₂).symm } | rfl | theorem | alg_equiv.symm_mk | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (alg_equiv.refl : A₁ ≃ₐ[R] A₁).symm = alg_equiv.refl | rfl | theorem | alg_equiv.refl_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_equiv_symm (f : A₁ ≃ₐ[R] A₁) : (f : A₁ ≃+* A₁).symm = f.symm | rfl | lemma | alg_equiv.to_ring_equiv_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_ring_equiv : (e.symm : A₂ ≃+* A₁) = (e : A₁ ≃+* A₂).symm | rfl | lemma | alg_equiv.symm_to_ring_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : A₁ ≃ₐ[R] A₃ | { commutes' := λ r, show e₂.to_fun (e₁.to_fun _) = _, by rw [e₁.commutes', e₂.commutes'],
..(e₁.to_ring_equiv.trans e₂.to_ring_equiv), } | def | alg_equiv.trans | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | Algebra equivalences are transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (e : A₁ ≃ₐ[R] A₂) : ∀ x, e (e.symm x) = x | e.to_equiv.apply_symm_apply | lemma | alg_equiv.apply_symm_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (e : A₁ ≃ₐ[R] A₂) : ∀ x, e.symm (e x) = x | e.to_equiv.symm_apply_apply | lemma | alg_equiv.symm_apply_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₃) :
(e₁.trans e₂).symm x = e₁.symm (e₂.symm x) | rfl | lemma | alg_equiv.symm_trans_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) :
⇑(e₁.trans e₂) = e₂ ∘ e₁ | rfl | lemma | alg_equiv.coe_trans | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₁) :
(e₁.trans e₂) x = e₂ (e₁ x) | rfl | lemma | alg_equiv.trans_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm (e : A₁ ≃ₐ[R] A₂) :
alg_hom.comp (e : A₁ →ₐ[R] A₂) ↑e.symm = alg_hom.id R A₂ | by { ext, simp } | lemma | alg_equiv.comp_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.comp",
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp (e : A₁ ≃ₐ[R] A₂) :
alg_hom.comp ↑e.symm (e : A₁ →ₐ[R] A₂) = alg_hom.id R A₁ | by { ext, simp } | lemma | alg_equiv.symm_comp | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.comp",
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_symm (e : A₁ ≃ₐ[R] A₂) : function.left_inverse e.symm e | e.left_inv | theorem | alg_equiv.left_inverse_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inverse_symm (e : A₁ ≃ₐ[R] A₂) : function.right_inverse e.symm e | e.right_inv | theorem | alg_equiv.right_inverse_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arrow_congr {A₁' A₂' : Type*} [semiring A₁'] [semiring A₂'] [algebra R A₁'] [algebra R A₂']
(e₁ : A₁ ≃ₐ[R] A₁') (e₂ : A₂ ≃ₐ[R] A₂') : (A₁ →ₐ[R] A₂) ≃ (A₁' →ₐ[R] A₂') | { to_fun := λ f, (e₂.to_alg_hom.comp f).comp e₁.symm.to_alg_hom,
inv_fun := λ f, (e₂.symm.to_alg_hom.comp f).comp e₁.to_alg_hom,
left_inv := λ f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, symm_comp],
simp only [←alg_hom.comp_assoc, symm_comp, alg_hom.id_comp, alg_hom.comp_id] },
right_inv := λ f,... | def | alg_equiv.arrow_congr | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.comp_assoc",
"alg_hom.comp_id",
"alg_hom.id_comp",
"algebra",
"inv_fun",
"semiring"
] | If `A₁` is equivalent to `A₁'` and `A₂` is equivalent to `A₂'`, then the type of maps
`A₁ →ₐ[R] A₂` is equivalent to the type of maps `A₁' →ₐ[R] A₂'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arrow_congr_comp {A₁' A₂' A₃' : Type*} [semiring A₁'] [semiring A₂'] [semiring A₃']
[algebra R A₁'] [algebra R A₂'] [algebra R A₃'] (e₁ : A₁ ≃ₐ[R] A₁') (e₂ : A₂ ≃ₐ[R] A₂')
(e₃ : A₃ ≃ₐ[R] A₃') (f : A₁ →ₐ[R] A₂) (g : A₂ →ₐ[R] A₃) :
arrow_congr e₁ e₃ (g.comp f) = (arrow_congr e₂ e₃ g).comp (arrow_congr e₁ e₂ f) | by { ext, simp only [arrow_congr, equiv.coe_fn_mk, alg_hom.comp_apply],
congr, exact (e₂.symm_apply_apply _).symm } | lemma | alg_equiv.arrow_congr_comp | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.comp_apply",
"algebra",
"equiv.coe_fn_mk",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arrow_congr_refl :
arrow_congr alg_equiv.refl alg_equiv.refl = equiv.refl (A₁ →ₐ[R] A₂) | by { ext, refl } | lemma | alg_equiv.arrow_congr_refl | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.refl",
"equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arrow_congr_trans {A₁' A₂' A₃' : Type*} [semiring A₁'] [semiring A₂'] [semiring A₃']
[algebra R A₁'] [algebra R A₂'] [algebra R A₃'] (e₁ : A₁ ≃ₐ[R] A₂) (e₁' : A₁' ≃ₐ[R] A₂')
(e₂ : A₂ ≃ₐ[R] A₃) (e₂' : A₂' ≃ₐ[R] A₃') :
arrow_congr (e₁.trans e₂) (e₁'.trans e₂') = (arrow_congr e₁ e₁').trans (arrow_congr e₂ e₂') | by { ext, refl } | lemma | alg_equiv.arrow_congr_trans | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"algebra",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arrow_congr_symm {A₁' A₂' : Type*} [semiring A₁'] [semiring A₂']
[algebra R A₁'] [algebra R A₂'] (e₁ : A₁ ≃ₐ[R] A₁') (e₂ : A₂ ≃ₐ[R] A₂') :
(arrow_congr e₁ e₂).symm = arrow_congr e₁.symm e₂.symm | by { ext, refl } | lemma | alg_equiv.arrow_congr_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"algebra",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_alg_hom (f : A₁ →ₐ[R] A₂) (g : A₂ →ₐ[R] A₁) (h₁ : f.comp g = alg_hom.id R A₂)
(h₂ : g.comp f = alg_hom.id R A₁) : A₁ ≃ₐ[R] A₂ | { to_fun := f,
inv_fun := g,
left_inv := alg_hom.ext_iff.1 h₂,
right_inv := alg_hom.ext_iff.1 h₁,
..f } | def | alg_equiv.of_alg_hom | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.id",
"inv_fun"
] | If an algebra morphism has an inverse, it is a algebra isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_alg_hom_of_alg_hom (f : A₁ →ₐ[R] A₂) (g : A₂ →ₐ[R] A₁) (h₁ h₂) :
↑(of_alg_hom f g h₁ h₂) = f | alg_hom.ext $ λ _, rfl | lemma | alg_equiv.coe_alg_hom_of_alg_hom | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_alg_hom_coe_alg_hom (f : A₁ ≃ₐ[R] A₂) (g : A₂ →ₐ[R] A₁) (h₁ h₂) :
of_alg_hom ↑f g h₁ h₂ = f | ext $ λ _, rfl | lemma | alg_equiv.of_alg_hom_coe_alg_hom | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_alg_hom_symm (f : A₁ →ₐ[R] A₂) (g : A₂ →ₐ[R] A₁) (h₁ h₂) :
(of_alg_hom f g h₁ h₂).symm = of_alg_hom g f h₂ h₁ | rfl | lemma | alg_equiv.of_alg_hom_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective (f : A₁ →ₐ[R] A₂) (hf : function.bijective f) : A₁ ≃ₐ[R] A₂ | { .. ring_equiv.of_bijective (f : A₁ →+* A₂) hf, .. f } | def | alg_equiv.of_bijective | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"ring_equiv.of_bijective"
] | Promotes a bijective algebra homomorphism to an algebra equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_bijective {f : A₁ →ₐ[R] A₂} {hf : function.bijective f} :
(alg_equiv.of_bijective f hf : A₁ → A₂) = f | rfl | lemma | alg_equiv.coe_of_bijective | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective_apply {f : A₁ →ₐ[R] A₂} {hf : function.bijective f} (a : A₁) :
(alg_equiv.of_bijective f hf) a = f a | rfl | lemma | alg_equiv.of_bijective_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv (e : A₁ ≃ₐ[R] A₂) : A₁ ≃ₗ[R] A₂ | { to_fun := e,
map_smul' := e.map_smul,
inv_fun := e.symm,
.. e } | def | alg_equiv.to_linear_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"inv_fun"
] | Forgetting the multiplicative structures, an equivalence of algebras is a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_linear_equiv_refl :
(alg_equiv.refl : A₁ ≃ₐ[R] A₁).to_linear_equiv = linear_equiv.refl R A₁ | rfl | lemma | alg_equiv.to_linear_equiv_refl | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.refl",
"linear_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_symm (e : A₁ ≃ₐ[R] A₂) :
e.to_linear_equiv.symm = e.symm.to_linear_equiv | rfl | lemma | alg_equiv.to_linear_equiv_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) :
(e₁.trans e₂).to_linear_equiv = e₁.to_linear_equiv.trans e₂.to_linear_equiv | rfl | lemma | alg_equiv.to_linear_equiv_trans | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_injective : function.injective (to_linear_equiv : _ → (A₁ ≃ₗ[R] A₂)) | λ e₁ e₂ h, ext $ linear_equiv.congr_fun h | theorem | alg_equiv.to_linear_equiv_injective | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"linear_equiv.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map : A₁ →ₗ[R] A₂ | e.to_alg_hom.to_linear_map | def | alg_equiv.to_linear_map | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | Interpret an algebra equivalence as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_alg_hom_to_linear_map :
(e : A₁ →ₐ[R] A₂).to_linear_map = e.to_linear_map | rfl | lemma | alg_equiv.to_alg_hom_to_linear_map | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_to_linear_map :
e.to_linear_equiv.to_linear_map = e.to_linear_map | rfl | lemma | alg_equiv.to_linear_equiv_to_linear_map | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_apply (x : A₁) : e.to_linear_map x = e x | rfl | lemma | alg_equiv.to_linear_map_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map_injective : function.injective (to_linear_map : _ → (A₁ →ₗ[R] A₂)) | λ e₁ e₂ h, ext $ linear_map.congr_fun h | theorem | alg_equiv.to_linear_map_injective | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"linear_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_to_linear_map (f : A₁ ≃ₐ[R] A₂) (g : A₂ ≃ₐ[R] A₃) :
(f.trans g).to_linear_map = g.to_linear_map.comp f.to_linear_map | rfl | lemma | alg_equiv.trans_to_linear_map | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_linear_equiv : A₁ ≃ₐ[R] A₂ | { to_fun := l,
inv_fun := l.symm,
map_mul' := map_mul,
commutes' := commutes,
..l } | def | alg_equiv.of_linear_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"inv_fun",
"map_mul"
] | Upgrade a linear equivalence to an algebra equivalence,
given that it distributes over multiplication and action of scalars. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_linear_equiv_symm :
(of_linear_equiv l map_mul commutes).symm = of_linear_equiv l.symm
((of_linear_equiv l map_mul commutes).symm.map_mul)
((of_linear_equiv l map_mul commutes).symm.commutes) | rfl | lemma | alg_equiv.of_linear_equiv_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_linear_equiv_to_linear_equiv (map_mul) (commutes) :
of_linear_equiv e.to_linear_equiv map_mul commutes = e | by { ext, refl } | lemma | alg_equiv.of_linear_equiv_to_linear_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_equiv_of_linear_equiv :
to_linear_equiv (of_linear_equiv l map_mul commutes) = l | by { ext, refl } | lemma | alg_equiv.to_linear_equiv_of_linear_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ring_equiv {f : A₁ ≃+* A₂}
(hf : ∀ x, f (algebra_map R A₁ x) = algebra_map R A₂ x) : A₁ ≃ₐ[R] A₂ | { to_fun := f,
inv_fun := f.symm,
commutes' := hf,
.. f } | def | alg_equiv.of_ring_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"algebra_map",
"inv_fun"
] | Promotes a linear ring_equiv to an alg_equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aut : group (A₁ ≃ₐ[R] A₁) | { mul := λ ϕ ψ, ψ.trans ϕ,
mul_assoc := λ ϕ ψ χ, rfl,
one := refl,
one_mul := λ ϕ, ext $ λ x, rfl,
mul_one := λ ϕ, ext $ λ x, rfl,
inv := symm,
mul_left_inv := λ ϕ, ext $ symm_apply_apply ϕ } | instance | alg_equiv.aut | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"group",
"mul_assoc",
"mul_left_inv",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : A₁) : (1 : A₁ ≃ₐ[R] A₁) x = x | rfl | lemma | alg_equiv.one_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (e₁ e₂ : A₁ ≃ₐ[R] A₁) (x : A₁) : (e₁ * e₂) x = e₁ (e₂ x) | rfl | lemma | alg_equiv.mul_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aut_congr (ϕ : A₁ ≃ₐ[R] A₂) : (A₁ ≃ₐ[R] A₁) ≃* (A₂ ≃ₐ[R] A₂) | { to_fun := λ ψ, ϕ.symm.trans (ψ.trans ϕ),
inv_fun := λ ψ, ϕ.trans (ψ.trans ϕ.symm),
left_inv := λ ψ, by { ext, simp_rw [trans_apply, symm_apply_apply] },
right_inv := λ ψ, by { ext, simp_rw [trans_apply, apply_symm_apply] },
map_mul' := λ ψ χ, by { ext, simp only [mul_apply, trans_apply, symm_apply_apply] } } | def | alg_equiv.aut_congr | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"inv_fun"
] | An algebra isomorphism induces a group isomorphism between automorphism groups | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aut_congr_refl : aut_congr (alg_equiv.refl) = mul_equiv.refl (A₁ ≃ₐ[R] A₁) | by { ext, refl } | lemma | alg_equiv.aut_congr_refl | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.refl",
"mul_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aut_congr_symm (ϕ : A₁ ≃ₐ[R] A₂) : (aut_congr ϕ).symm = aut_congr ϕ.symm | rfl | lemma | alg_equiv.aut_congr_symm | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aut_congr_trans (ϕ : A₁ ≃ₐ[R] A₂) (ψ : A₂ ≃ₐ[R] A₃) :
(aut_congr ϕ).trans (aut_congr ψ) = aut_congr (ϕ.trans ψ) | rfl | lemma | alg_equiv.aut_congr_trans | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_mul_semiring_action : mul_semiring_action (A₁ ≃ₐ[R] A₁) A₁ | { smul := ($),
smul_zero := alg_equiv.map_zero,
smul_add := alg_equiv.map_add,
smul_one := alg_equiv.map_one,
smul_mul := alg_equiv.map_mul,
one_smul := λ _, rfl,
mul_smul := λ _ _ _, rfl } | instance | alg_equiv.apply_mul_semiring_action | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"alg_equiv.map_add",
"alg_equiv.map_mul",
"alg_equiv.map_one",
"alg_equiv.map_zero",
"mul_semiring_action",
"one_smul",
"smul_add",
"smul_zero"
] | The tautological action by `A₁ ≃ₐ[R] A₁` on `A₁`.
This generalizes `function.End.apply_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (f : A₁ ≃ₐ[R] A₁) (a : A₁) : f • a = f a | rfl | lemma | alg_equiv.smul_def | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul : has_faithful_smul (A₁ ≃ₐ[R] A₁) A₁ | ⟨λ _ _, alg_equiv.ext⟩ | instance | alg_equiv.apply_has_faithful_smul | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"has_faithful_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_smul_comm_class : smul_comm_class R (A₁ ≃ₐ[R] A₁) A₁ | { smul_comm := λ r e a, (e.map_smul r a).symm } | instance | alg_equiv.apply_smul_comm_class | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_smul_comm_class' : smul_comm_class (A₁ ≃ₐ[R] A₁) R A₁ | { smul_comm := λ e r a, (e.map_smul r a) } | instance | alg_equiv.apply_smul_comm_class' | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_apply (e : A₁ ≃ₐ[R] A₂) {y : R} {x : A₁} :
(algebra_map R A₂ y = e x) ↔ (algebra_map R A₁ y = x) | ⟨λ h, by simpa using e.symm.to_alg_hom.algebra_map_eq_apply h,
λ h, e.to_alg_hom.algebra_map_eq_apply h⟩ | lemma | alg_equiv.algebra_map_eq_apply | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod {ι : Type*} (f : ι → A₁) (s : finset ι) :
e (∏ x in s, f x) = ∏ x in s, e (f x) | map_prod _ f s | lemma | alg_equiv.map_prod | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_finsupp_prod {α : Type*} [has_zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A₁) :
e (f.prod g) = f.prod (λ i a, e (g i a)) | map_finsupp_prod _ f g | lemma | alg_equiv.map_finsupp_prod | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"map_finsupp_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x) : e (-x) = -e x | map_neg e x | lemma | alg_equiv.map_neg | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y) : e (x - y) = e x - e y | map_sub e x y | lemma | alg_equiv.map_sub | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_alg_equiv (g : G) : A ≃ₐ[R] A | { .. mul_semiring_action.to_ring_equiv _ _ g,
.. mul_semiring_action.to_alg_hom R A g } | def | mul_semiring_action.to_alg_equiv | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"mul_semiring_action.to_alg_hom",
"mul_semiring_action.to_ring_equiv"
] | Each element of the group defines a algebra equivalence.
This is a stronger version of `mul_semiring_action.to_ring_equiv` and
`distrib_mul_action.to_linear_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_alg_equiv_injective [has_faithful_smul G A] :
function.injective (mul_semiring_action.to_alg_equiv R A : G → A ≃ₐ[R] A) | λ m₁ m₂ h, eq_of_smul_eq_smul $ λ r, alg_equiv.ext_iff.1 h r | theorem | mul_semiring_action.to_alg_equiv_injective | algebra.algebra | src/algebra/algebra/equiv.lean | [
"algebra.algebra.hom"
] | [
"has_faithful_smul",
"mul_semiring_action.to_alg_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom (R : Type u) (A : Type v) (B : Type w)
[comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] extends ring_hom A B | (commutes' : ∀ r : R, to_fun (algebra_map R A r) = algebra_map R B r) | structure | alg_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"ring_hom",
"semiring"
] | Defining the homomorphism in the category R-Alg. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_class (F : Type*) (R : out_param Type*) (A : out_param Type*) (B : out_param Type*)
[comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B]
extends ring_hom_class F A B | (commutes : ∀ (f : F) (r : R), f (algebra_map R A r) = algebra_map R B r) | class | alg_hom_class | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"ring_hom_class",
"semiring"
] | `alg_hom_class F R A B` asserts `F` is a type of bundled algebra homomorphisms
from `A` to `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe {F : Type*} [alg_hom_class F R A B] (f : F) :
⇑(f : A →ₐ[R] B) = f | rfl | lemma | alg_hom.coe_coe | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"alg_hom_class",
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : A →ₐ[R] B) : f.to_fun = f | rfl | lemma | alg_hom.to_fun_eq_coe | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom : has_coe (A →ₐ[R] B) (A →+* B) | ⟨alg_hom.to_ring_hom⟩ | instance | alg_hom.coe_ring_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom : has_coe (A →ₐ[R] B) (A →* B) | ⟨λ f, ↑(f : A →+* B)⟩ | instance | alg_hom.coe_monoid_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom : has_coe (A →ₐ[R] B) (A →+ B) | ⟨λ f, ↑(f : A →+* B)⟩ | instance | alg_hom.coe_add_monoid_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk {f : A → B} (h₁ h₂ h₃ h₄ h₅) :
⇑(⟨f, h₁, h₂, h₃, h₄, h₅⟩ : A →ₐ[R] B) = f | rfl | lemma | alg_hom.coe_mk | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_eq_coe (f : A →ₐ[R] B) : f.to_ring_hom = f | rfl | lemma | alg_hom.to_ring_hom_eq_coe | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_ring_hom (f : A →ₐ[R] B) : ⇑(f : A →+* B) = f | rfl | lemma | alg_hom.coe_to_ring_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_monoid_hom (f : A →ₐ[R] B) : ⇑(f : A →* B) = f | rfl | lemma | alg_hom.coe_to_monoid_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_monoid_hom (f : A →ₐ[R] B) : ⇑(f : A →+ B) = f | rfl | lemma | alg_hom.coe_to_add_monoid_hom | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_injective : @function.injective (A →ₐ[R] B) (A → B) coe_fn | fun_like.coe_injective | theorem | alg_hom.coe_fn_injective | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_inj {φ₁ φ₂ : A →ₐ[R] B} : (φ₁ : A → B) = φ₂ ↔ φ₁ = φ₂ | fun_like.coe_fn_eq | theorem | alg_hom.coe_fn_inj | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"fun_like.coe_fn_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom_injective : function.injective (coe : (A →ₐ[R] B) → (A →+* B)) | λ φ₁ φ₂ H, coe_fn_injective $ show ((φ₁ : (A →+* B)) : A → B) = ((φ₂ : (A →+* B)) : A → B),
from congr_arg _ H | theorem | alg_hom.coe_ring_hom_injective | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_injective : function.injective (coe : (A →ₐ[R] B) → (A →* B)) | ring_hom.coe_monoid_hom_injective.comp coe_ring_hom_injective | theorem | alg_hom.coe_monoid_hom_injective | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_injective : function.injective (coe : (A →ₐ[R] B) → (A →+ B)) | ring_hom.coe_add_monoid_hom_injective.comp coe_ring_hom_injective | theorem | alg_hom.coe_add_monoid_hom_injective | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x | fun_like.congr_fun H x | lemma | alg_hom.congr_fun | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y | fun_like.congr_arg φ h | lemma | alg_hom.congr_arg | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"fun_like.congr_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ x, φ₁ x = φ₂ x) : φ₁ = φ₂ | fun_like.ext _ _ H | theorem | alg_hom.ext | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ x, φ₁ x = φ₂ x | fun_like.ext_iff | theorem | alg_hom.ext_iff | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe {f : A →ₐ[R] B} (h₁ h₂ h₃ h₄ h₅) :
(⟨f, h₁, h₂, h₃, h₄, h₅⟩ : A →ₐ[R] B) = f | ext $ λ _, rfl | theorem | alg_hom.mk_coe | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutes (r : R) : φ (algebra_map R A r) = algebra_map R B r | φ.commutes' r | theorem | alg_hom.commutes | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_algebra_map : (φ : A →+* B).comp (algebra_map R A) = algebra_map R B | ring_hom.ext $ φ.commutes | theorem | alg_hom.comp_algebra_map | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"algebra_map",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (r s : A) : φ (r + s) = φ r + φ s | map_add _ _ _ | lemma | alg_hom.map_add | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : φ 0 = 0 | map_zero _ | lemma | alg_hom.map_zero | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (x y) : φ (x * y) = φ x * φ y | map_mul _ _ _ | lemma | alg_hom.map_mul | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one : φ 1 = 1 | map_one _ | lemma | alg_hom.map_one | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pow (x : A) (n : ℕ) : φ (x ^ n) = (φ x) ^ n | map_pow _ _ _ | lemma | alg_hom.map_pow | algebra.algebra | src/algebra/algebra/hom.lean | [
"algebra.algebra.basic"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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