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
value |
|---|---|---|---|---|---|---|---|---|---|---|
coe_injective :
@function.injective (A →ₙₐ[R] B) (A → B) coe_fn | by rintro ⟨f, _⟩ ⟨g, _⟩ ⟨h⟩; congr | lemma | non_unital_alg_hom.coe_injective | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : A →ₙₐ[R] B} (h : ∀ x, f x = g x) : f = g | coe_injective $ funext h | lemma | non_unital_alg_hom.ext | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : A →ₙₐ[R] B} : f = g ↔ ∀ x, f x = g x | ⟨by { rintro rfl x, refl }, ext⟩ | lemma | non_unital_alg_hom.ext_iff | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : A →ₙₐ[R] B} (h : f = g) (x : A) : f x = g x | h ▸ rfl | lemma | non_unital_alg_hom.congr_fun | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : A → B) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : A →ₙₐ[R] B) : A → B) = f | rfl | lemma | non_unital_alg_hom.coe_mk | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : A →ₙₐ[R] B) (h₁ h₂ h₃ h₄) :
(⟨f, h₁, h₂, h₃, h₄⟩ : A →ₙₐ[R] B) = f | by { ext, refl, } | lemma | non_unital_alg_hom.mk_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_distrib_mul_action_hom_eq_coe (f : A →ₙₐ[R] B) :
f.to_distrib_mul_action_hom = ↑f | rfl | lemma | non_unital_alg_hom.to_distrib_mul_action_hom_eq_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_hom_eq_coe (f : A →ₙₐ[R] B) : f.to_mul_hom = ↑f | rfl | lemma | non_unital_alg_hom.to_mul_hom_eq_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_distrib_mul_action_hom (f : A →ₙₐ[R] B) :
((f : A →+[R] B) : A → B) = f | rfl | lemma | non_unital_alg_hom.coe_to_distrib_mul_action_hom | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mul_hom (f : A →ₙₐ[R] B) :
((f : A →ₙ* B) : A → B) = f | rfl | lemma | non_unital_alg_hom.coe_to_mul_hom | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_distrib_mul_action_hom_injective {f g : A →ₙₐ[R] B}
(h : (f : A →+[R] B) = (g : A →+[R] B)) : f = g | by { ext a, exact distrib_mul_action_hom.congr_fun h a, } | lemma | non_unital_alg_hom.to_distrib_mul_action_hom_injective | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"distrib_mul_action_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_hom_injective {f g : A →ₙₐ[R] B}
(h : (f : A →ₙ* B) = (g : A →ₙ* B)) : f = g | by { ext a, exact mul_hom.congr_fun h a, } | lemma | non_unital_alg_hom.to_mul_hom_injective | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"mul_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_distrib_mul_action_hom_mk (f : A →ₙₐ[R] B) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : A →ₙₐ[R] B) : A →+[R] B) =
⟨f, h₁, h₂, h₃⟩ | by { ext, refl, } | lemma | non_unital_alg_hom.coe_distrib_mul_action_hom_mk | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_hom_mk (f : A →ₙₐ[R] B) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : A →ₙₐ[R] B) : A →ₙ* B) = ⟨f, h₄⟩ | by { ext, refl, } | lemma | non_unital_alg_hom.coe_mul_hom_mk | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (f : A →ₙₐ[R] B) (c : R) (x : A) :
f (c • x) = c • f x | map_smul _ _ _ | lemma | non_unital_alg_hom.map_smul | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (f : A →ₙₐ[R] B) (x y : A) :
f (x + y) = (f x) + (f y) | map_add _ _ _ | lemma | non_unital_alg_hom.map_add | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (f : A →ₙₐ[R] B) (x y : A) :
f (x * y) = (f x) * (f y) | map_mul _ _ _ | lemma | non_unital_alg_hom.map_mul | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero (f : A →ₙₐ[R] B) : f 0 = 0 | map_zero _ | lemma | non_unital_alg_hom.map_zero | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : A →ₙₐ[R] B) : A → B) = 0 | rfl | lemma | non_unital_alg_hom.coe_zero | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : A →ₙₐ[R] A) : A → A) = id | rfl | lemma | non_unital_alg_hom.coe_one | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (a : A) : (0 : A →ₙₐ[R] B) a = 0 | rfl | lemma | non_unital_alg_hom.zero_apply | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (a : A) : (1 : A →ₙₐ[R] A) a = a | rfl | lemma | non_unital_alg_hom.one_apply | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : B →ₙₐ[R] C) (g : A →ₙₐ[R] B) : A →ₙₐ[R] C | { .. (f : B →ₙ* C).comp (g : A →ₙ* B),
.. (f : B →+[R] C).comp (g : A →+[R] B) } | def | non_unital_alg_hom.comp | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | The composition of morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : B →ₙₐ[R] C) (g : A →ₙₐ[R] B) :
(f.comp g : A → C) = (f : B → C) ∘ (g : A → B) | rfl | lemma | non_unital_alg_hom.coe_comp | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : B →ₙₐ[R] C) (g : A →ₙₐ[R] B) (x : A) :
f.comp g x = f (g x) | rfl | lemma | non_unital_alg_hom.comp_apply | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse (f : A →ₙₐ[R] B) (g : B → A)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) :
B →ₙₐ[R] A | { .. (f : A →ₙ* B).inverse g h₁ h₂,
.. (f : A →+[R] B).inverse g h₁ h₂ } | def | non_unital_alg_hom.inverse | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | The inverse of a bijective morphism is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inverse (f : A →ₙₐ[R] B) (g : B → A)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) :
(inverse f g h₁ h₂ : B → A) = g | rfl | lemma | non_unital_alg_hom.coe_inverse | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : A × B →ₙₐ[R] A | { to_fun := prod.fst,
map_zero' := rfl, map_add' := λ x y, rfl, map_smul' := λ x y, rfl, map_mul' := λ x y, rfl } | def | non_unital_alg_hom.fst | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | The first projection of a product is a non-unital alg_hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : A × B →ₙₐ[R] B | { to_fun := prod.snd,
map_zero' := rfl, map_add' := λ x y, rfl, map_smul' := λ x y, rfl, map_mul' := λ x y, rfl } | def | non_unital_alg_hom.snd | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | The second projection of a product is a non-unital alg_hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : A →ₙₐ[R] B) (g : A →ₙₐ[R] C) : (A →ₙₐ[R] B × C) | { to_fun := pi.prod f g,
map_zero' := by simp only [pi.prod, prod.zero_eq_mk, map_zero],
map_add' := λ x y, by simp only [pi.prod, prod.mk_add_mk, map_add],
map_mul' := λ x y, by simp only [pi.prod, prod.mk_mul_mk, map_mul],
map_smul' := λ c x, by simp only [pi.prod, prod.smul_mk, map_smul, ring_hom.id_app... | def | non_unital_alg_hom.prod | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"map_mul",
"pi.prod",
"prod.mk_mul_mk",
"prod.smul_mk",
"ring_hom.id_apply"
] | The prod of two morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (f : A →ₙₐ[R] B) (g : A →ₙₐ[R] C) : ⇑(f.prod g) = pi.prod f g | rfl | lemma | non_unital_alg_hom.coe_prod | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"pi.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_prod (f : A →ₙₐ[R] B) (g : A →ₙₐ[R] C) :
(fst R B C).comp (prod f g) = f | by ext; refl | theorem | non_unital_alg_hom.fst_prod | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_prod (f : A →ₙₐ[R] B) (g : A →ₙₐ[R] C) :
(snd R B C).comp (prod f g) = g | by ext; refl | theorem | non_unital_alg_hom.snd_prod | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_fst_snd : prod (fst R A B) (snd R A B) = 1 | coe_injective pi.prod_fst_snd | theorem | non_unital_alg_hom.prod_fst_snd | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"pi.prod_fst_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv : ((A →ₙₐ[R] B) × (A →ₙₐ[R] C)) ≃ (A →ₙₐ[R] B × C) | { to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((fst _ _ _).comp f, (snd _ _ _).comp f),
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl } | def | non_unital_alg_hom.prod_equiv | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"inv_fun"
] | Taking the product of two maps with the same domain is equivalent to taking the product of
their codomains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl : A →ₙₐ[R] A × B | prod 1 0 | def | non_unital_alg_hom.inl | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | The left injection into a product is a non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr : B →ₙₐ[R] A × B | prod 0 1 | def | non_unital_alg_hom.inr | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | The right injection into a product is a non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inl : (inl R A B : A → A × B) = λ x, (x, 0) | rfl | theorem | non_unital_alg_hom.coe_inl | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_apply (x : A) : inl R A B x = (x, 0) | rfl | theorem | non_unital_alg_hom.inl_apply | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inr : (inr R A B : B → A × B) = prod.mk 0 | rfl | theorem | non_unital_alg_hom.coe_inr | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_apply (x : B) : inr R A B x = (0, x) | rfl | theorem | non_unital_alg_hom.inr_apply | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_alg_hom (f : A →ₐ[R] B) : A →ₙₐ[R] B | { map_smul' := map_smul f, .. f, } | def | alg_hom.to_non_unital_alg_hom | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | A unital morphism of algebras is a `non_unital_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_alg_hom.has_coe : has_coe (A →ₐ[R] B) (A →ₙₐ[R] B) | ⟨to_non_unital_alg_hom⟩ | instance | alg_hom.non_unital_alg_hom.has_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_alg_hom_eq_coe (f : A →ₐ[R] B) : f.to_non_unital_alg_hom = f | rfl | lemma | alg_hom.to_non_unital_alg_hom_eq_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_non_unital_alg_hom (f : A →ₐ[R] B) :
((f : A →ₙₐ[R] B) : A → B) = f | rfl | lemma | alg_hom.coe_to_non_unital_alg_hom | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_ring_hom (α β : Type*) [non_unital_non_assoc_semiring α]
[non_unital_non_assoc_semiring β] extends α →ₙ* β, α →+ β | structure | non_unital_ring_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_non_assoc_semiring"
] | Bundled non-unital semiring homomorphisms `α →ₙ+* β`; use this for bundled non-unital ring
homomorphisms too.
When possible, instead of parametrizing results over `(f : α →ₙ+* β)`,
you should parametrize over `(F : Type*) [non_unital_ring_hom_class F α β] (f : F)`.
When you extend this structure, make sure to extend ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_ring_hom_class (F : Type*) (α β : out_param Type*)
[non_unital_non_assoc_semiring α] [non_unital_non_assoc_semiring β]
extends mul_hom_class F α β, add_monoid_hom_class F α β | class | non_unital_ring_hom_class | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"add_monoid_hom_class",
"mul_hom_class",
"non_unital_non_assoc_semiring"
] | `non_unital_ring_hom_class F α β` states that `F` is a type of non-unital (semi)ring
homomorphisms. You should extend this class when you extend `non_unital_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : α →ₙ+* β) : f.to_fun = f | rfl | lemma | non_unital_ring_hom.to_fun_eq_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α → β) (h₁ h₂ h₃) : ⇑(⟨f, h₁, h₂, h₃⟩ : α →ₙ+* β) = f | rfl | lemma | non_unital_ring_hom.coe_mk | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe [non_unital_ring_hom_class F α β] (f : F) : ((f : α →ₙ+* β) : α → β) = f | rfl | lemma | non_unital_ring_hom.coe_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"coe_coe",
"non_unital_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mul_hom (f : α →ₙ+* β) : ⇑f.to_mul_hom = f | rfl | lemma | non_unital_ring_hom.coe_to_mul_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_hom_mk (f : α → β) (h₁ h₂ h₃) :
((⟨f, h₁, h₂, h₃⟩ : α →ₙ+* β) : α →ₙ* β) = ⟨f, h₁⟩ | rfl | lemma | non_unital_ring_hom.coe_mul_hom_mk | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_add_monoid_hom (f : α →ₙ+* β) : ⇑f.to_add_monoid_hom = f | rfl | lemma | non_unital_ring_hom.coe_to_add_monoid_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_mk (f : α → β) (h₁ h₂ h₃) :
((⟨f, h₁, h₂, h₃⟩ : α →ₙ+* β) : α →+ β) = ⟨f, h₂, h₃⟩ | rfl | lemma | non_unital_ring_hom.coe_add_monoid_hom_mk | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : α →ₙ+* β | { ..f.to_mul_hom.copy f' h, ..f.to_add_monoid_hom.copy f' h } | def | non_unital_ring_hom.copy | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Copy of a `ring_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | non_unital_ring_hom.coe_copy | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →ₙ+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | non_unital_ring_hom.copy_eq | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g | fun_like.ext _ _ | lemma | non_unital_ring_hom.ext | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : α →ₙ+* β} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | non_unital_ring_hom.ext_iff | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : α →ₙ+* β) (h₁ h₂ h₃) : non_unital_ring_hom.mk f h₁ h₂ h₃ = f | ext $ λ _, rfl | lemma | non_unital_ring_hom.mk_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_injective : injective (coe : (α →ₙ+* β) → (α →+ β)) | λ f g h, ext $ add_monoid_hom.congr_fun h | lemma | non_unital_ring_hom.coe_add_monoid_hom_injective | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_hom_injective : injective (coe : (α →ₙ+* β) → (α →ₙ* β)) | λ f g h, ext $ mul_hom.congr_fun h | lemma | non_unital_ring_hom.coe_mul_hom_injective | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"mul_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (α : Type*) [non_unital_non_assoc_semiring α] : α →ₙ+* α | by refine {to_fun := id, ..}; intros; refl | def | non_unital_ring_hom.id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_non_assoc_semiring"
] | The identity non-unital ring homomorphism from a non-unital semiring to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero : ⇑(0 : α →ₙ+* β) = 0 | rfl | lemma | non_unital_ring_hom.coe_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : α) : (0 : α →ₙ+* β) x = 0 | rfl | lemma | non_unital_ring_hom.zero_apply | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (x : α) : non_unital_ring_hom.id α x = x | rfl | lemma | non_unital_ring_hom.id_apply | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_id :
(non_unital_ring_hom.id α : α →+ α) = add_monoid_hom.id α | rfl | lemma | non_unital_ring_hom.coe_add_monoid_hom_id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_hom_id : (non_unital_ring_hom.id α : α →ₙ* α) = mul_hom.id α | rfl | lemma | non_unital_ring_hom.coe_mul_hom_id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"mul_hom.id",
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : α →ₙ+* γ | { ..g.to_mul_hom.comp f.to_mul_hom, ..g.to_add_monoid_hom.comp f.to_add_monoid_hom } | def | non_unital_ring_hom.comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Composition of non-unital ring homomorphisms is a non-unital ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_assoc {δ} {rδ : non_unital_non_assoc_semiring δ} (f : α →ₙ+* β) (g : β →ₙ+* γ)
(h : γ →ₙ+* δ) : (h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | non_unital_ring_hom.comp_assoc | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_non_assoc_semiring"
] | Composition of non-unital ring homomorphisms is associative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g ∘ f | rfl | lemma | non_unital_ring_hom.coe_comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) | rfl | lemma | non_unital_ring_hom.comp_apply | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_add_monoid_hom (g : β →ₙ+* γ) (f : α →ₙ+* β) :
(g.comp f : α →+ γ) = (g : β →+ γ).comp f | rfl | lemma | non_unital_ring_hom.coe_comp_add_monoid_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_mul_hom (g : β →ₙ+* γ) (f : α →ₙ+* β) :
(g.comp f : α →ₙ* γ) = (g : β →ₙ* γ).comp f | rfl | lemma | non_unital_ring_hom.coe_comp_mul_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_zero (g : β →ₙ+* γ) : g.comp (0 : α →ₙ+* β) = 0 | by { ext, simp } | lemma | non_unital_ring_hom.comp_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp (f : α →ₙ+* β) : (0 : β →ₙ+* γ).comp f = 0 | by { ext, refl } | lemma | non_unital_ring_hom.zero_comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : α →ₙ+* β) : f.comp (non_unital_ring_hom.id α) = f | ext $ λ x, rfl | lemma | non_unital_ring_hom.comp_id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →ₙ+* β) : (non_unital_ring_hom.id β).comp f = f | ext $ λ x, rfl | lemma | non_unital_ring_hom.id_comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : α →ₙ+* α) = non_unital_ring_hom.id α | rfl | lemma | non_unital_ring_hom.one_def | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : α →ₙ+* α) = id | rfl | lemma | non_unital_ring_hom.coe_one | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (f g : α →ₙ+* α) : f * g = f.comp g | rfl | lemma | non_unital_ring_hom.mul_def | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : α →ₙ+* α) : ⇑(f * g) = f ∘ g | rfl | lemma | non_unital_ring_hom.coe_mul | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : β →ₙ+* γ} {f : α →ₙ+* β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 (ext_iff.1 h), λ h, h ▸ rfl⟩ | lemma | non_unital_ring_hom.cancel_right | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ x, hg $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩ | lemma | non_unital_ring_hom.cancel_left | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom (α : Type*) (β : Type*) [non_assoc_semiring α] [non_assoc_semiring β]
extends α →* β, α →+ β, α →ₙ+* β, α →*₀ β | structure | ring_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_assoc_semiring"
] | Bundled semiring homomorphisms; use this for bundled ring homomorphisms too.
This extends from both `monoid_hom` and `monoid_with_zero_hom` in order to put the fields in a
sensible order, even though `monoid_with_zero_hom` already extends `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_class (F : Type*) (α β : out_param Type*)
[non_assoc_semiring α] [non_assoc_semiring β]
extends monoid_hom_class F α β, add_monoid_hom_class F α β, monoid_with_zero_hom_class F α β | class | ring_hom_class | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"add_monoid_hom_class",
"monoid_hom_class",
"monoid_with_zero_hom_class",
"non_assoc_semiring"
] | `ring_hom_class F α β` states that `F` is a type of (semi)ring homomorphisms.
You should extend this class when you extend `ring_hom`.
This extends from both `monoid_hom_class` and `monoid_with_zero_hom_class` in
order to put the fields in a sensible order, even though
`monoid_with_zero_hom_class` already extends `mon... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bit1 (f : F) (a : α) : (f (bit1 a) : β) = bit1 (f a) | by simp [bit1] | lemma | map_bit1 | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Ring homomorphisms preserve `bit1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_class.to_non_unital_ring_hom_class : non_unital_ring_hom_class F α β | { .. ‹ring_hom_class F α β› } | instance | ring_hom_class.to_non_unital_ring_hom_class | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_unital_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : α →+* β) : f.to_fun = f | rfl | lemma | ring_hom.to_fun_eq_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α → β) (h₁ h₂ h₃ h₄) : ⇑(⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) = f | rfl | lemma | ring_hom.coe_mk | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe {F : Type*} [ring_hom_class F α β] (f : F) : ((f : α →+* β) : α → β) = f | rfl | lemma | ring_hom.coe_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"coe_coe",
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_monoid_hom : has_coe (α →+* β) (α →* β) | ⟨ring_hom.to_monoid_hom⟩ | instance | ring_hom.has_coe_monoid_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom (f : α →+* β) : ⇑(f : α →* β) = f | rfl | lemma | ring_hom.coe_monoid_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom_eq_coe (f : α →+* β) : f.to_monoid_hom = f | rfl | lemma | ring_hom.to_monoid_hom_eq_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_with_zero_hom_eq_coe (f : α →+* β) :
(f.to_monoid_with_zero_hom : α → β) = f | rfl | lemma | ring_hom.to_monoid_with_zero_hom_eq_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_mk (f : α → β) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) : α →* β) = ⟨f, h₁, h₂⟩ | rfl | lemma | ring_hom.coe_monoid_hom_mk | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom (f : α →+* β) : ⇑(f : α →+ β) = f | rfl | lemma | ring_hom.coe_add_monoid_hom | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom_eq_coe (f : α →+* β) : f.to_add_monoid_hom = f | rfl | lemma | ring_hom.to_add_monoid_hom_eq_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_mk (f : α → β) (h₁ h₂ h₃ h₄) :
((⟨f, h₁, h₂, h₃, h₄⟩ : α →+* β) : α →+ β) = ⟨f, h₃, h₄⟩ | rfl | lemma | ring_hom.coe_add_monoid_hom_mk | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →+* β) (f' : α → β) (h : f' = f) : α →+* β | { ..f.to_monoid_with_zero_hom.copy f' h, ..f.to_add_monoid_hom.copy f' h } | def | ring_hom.copy | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Copy of a `ring_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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