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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|---|---|---|---|---|---|---|---|---|---|---|
map_neg_one : f (-1) = -1 | f.map_one ▸ f.map_neg 1 | lemma | ring_equiv.map_neg_one | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_neg_one_iff {x : R} : f x = -1 ↔ x = -1 | by rw [← neg_eq_iff_eq_neg, ← neg_eq_iff_eq_neg, ← map_neg, ring_equiv.map_eq_one_iff] | lemma | ring_equiv.map_eq_neg_one_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.map_eq_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom (e : R ≃+* S) : R →ₙ+* S | { .. e.to_mul_equiv.to_mul_hom, .. e.to_add_equiv.to_add_monoid_hom } | def | ring_equiv.to_non_unital_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | Reinterpret a ring equivalence as a non-unital ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_non_unital_ring_hom_injective :
function.injective (to_non_unital_ring_hom : (R ≃+* S) → R →ₙ+* S) | λ f g h, ring_equiv.ext (non_unital_ring_hom.ext_iff.1 h) | lemma | ring_equiv.to_non_unital_ring_hom_injective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_non_unital_ring_hom : has_coe (R ≃+* S) (R →ₙ+* S) | ⟨ring_equiv.to_non_unital_ring_hom⟩ | instance | ring_equiv.has_coe_to_non_unital_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_eq_coe (f : R ≃+* S) : f.to_non_unital_ring_hom = ↑f | rfl | lemma | ring_equiv.to_non_unital_ring_hom_eq_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_non_unital_ring_hom (f : R ≃+* S) : ⇑(f : R →ₙ+* S) = f | rfl | lemma | ring_equiv.coe_to_non_unital_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_non_unital_ring_hom_inj_iff {R S : Type*}
[non_unital_non_assoc_semiring R] [non_unital_non_assoc_semiring S]
(f g : R ≃+* S) :
f = g ↔ (f : R →ₙ+* S) = g | ⟨congr_arg _, λ h, ext $ non_unital_ring_hom.ext_iff.mp h⟩ | lemma | ring_equiv.coe_non_unital_ring_hom_inj_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_refl :
(ring_equiv.refl R).to_non_unital_ring_hom = non_unital_ring_hom.id R | rfl | lemma | ring_equiv.to_non_unital_ring_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_ring_hom.id",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_apply_symm_to_non_unital_ring_hom_apply (e : R ≃+* S) :
∀ (y : S), e.to_non_unital_ring_hom (e.symm.to_non_unital_ring_hom y) = y | e.to_equiv.apply_symm_apply | lemma | ring_equiv.to_non_unital_ring_hom_apply_symm_to_non_unital_ring_hom_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_non_unital_ring_hom_apply_to_non_unital_ring_hom_apply (e : R ≃+* S) :
∀ (x : R), e.symm.to_non_unital_ring_hom (e.to_non_unital_ring_hom x) = x | equiv.symm_apply_apply (e.to_equiv) | lemma | ring_equiv.symm_to_non_unital_ring_hom_apply_to_non_unital_ring_hom_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂).to_non_unital_ring_hom = e₂.to_non_unital_ring_hom.comp e₁.to_non_unital_ring_hom | rfl | lemma | ring_equiv.to_non_unital_ring_hom_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_comp_symm_to_non_unital_ring_hom (e : R ≃+* S) :
e.to_non_unital_ring_hom.comp e.symm.to_non_unital_ring_hom = non_unital_ring_hom.id _ | by { ext, simp } | lemma | ring_equiv.to_non_unital_ring_hom_comp_symm_to_non_unital_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_non_unital_ring_hom_comp_to_non_unital_ring_hom (e : R ≃+* S) :
e.symm.to_non_unital_ring_hom.comp e.to_non_unital_ring_hom = non_unital_ring_hom.id _ | by { ext, simp } | lemma | ring_equiv.symm_to_non_unital_ring_hom_comp_to_non_unital_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_unital_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom (e : R ≃+* S) : R →+* S | { .. e.to_mul_equiv.to_monoid_hom, .. e.to_add_equiv.to_add_monoid_hom } | def | ring_equiv.to_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | Reinterpret a ring equivalence as a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ring_hom_injective : function.injective (to_ring_hom : (R ≃+* S) → R →+* S) | λ f g h, ring_equiv.ext (ring_hom.ext_iff.1 h) | lemma | ring_equiv.to_ring_hom_injective | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.ext",
"to_ring_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe_to_ring_hom : has_coe (R ≃+* S) (R →+* S) | ⟨ring_equiv.to_ring_hom⟩ | instance | ring_equiv.has_coe_to_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_eq_coe (f : R ≃+* S) : f.to_ring_hom = ↑f | rfl | lemma | ring_equiv.to_ring_hom_eq_coe | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_ring_hom (f : R ≃+* S) : ⇑(f : R →+* S) = f | rfl | lemma | ring_equiv.coe_to_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom_inj_iff {R S : Type*} [non_assoc_semiring R] [non_assoc_semiring S]
(f g : R ≃+* S) :
f = g ↔ (f : R →+* S) = g | ⟨congr_arg _, λ h, ext $ ring_hom.ext_iff.mp h⟩ | lemma | ring_equiv.coe_ring_hom_inj_iff | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_ring_hom_commutes (f : R ≃+* S) :
((f : R →+* S) : R →ₙ+* S) = (f : R →ₙ+* S) | rfl | lemma | ring_equiv.to_non_unital_ring_hom_commutes | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | The two paths coercion can take to a `non_unital_ring_hom` are equivalent | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_monoid_hom (e : R ≃+* S) : R →* S | e.to_ring_hom.to_monoid_hom | abbreviation | ring_equiv.to_monoid_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | Reinterpret a ring equivalence as a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_monoid_hom (e : R ≃+* S) : R →+ S | e.to_ring_hom.to_add_monoid_hom | abbreviation | ring_equiv.to_add_monoid_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | Reinterpret a ring equivalence as an `add_monoid` homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_monoid_hom_commutes (f : R ≃+* S) :
(f : R →+* S).to_add_monoid_hom = (f : R ≃+ S).to_add_monoid_hom | rfl | lemma | ring_equiv.to_add_monoid_hom_commutes | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | The two paths coercion can take to an `add_monoid_hom` are equivalent | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_monoid_hom_commutes (f : R ≃+* S) :
(f : R →+* S).to_monoid_hom = (f : R ≃* S).to_monoid_hom | rfl | lemma | ring_equiv.to_monoid_hom_commutes | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | The two paths coercion can take to an `monoid_hom` are equivalent | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_equiv_commutes (f : R ≃+* S) :
(f : R ≃+ S).to_equiv = (f : R ≃* S).to_equiv | rfl | lemma | ring_equiv.to_equiv_commutes | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | The two paths coercion can take to an `equiv` are equivalent | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ring_hom_refl : (ring_equiv.refl R).to_ring_hom = ring_hom.id R | rfl | lemma | ring_equiv.to_ring_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom_refl : (ring_equiv.refl R).to_monoid_hom = monoid_hom.id R | rfl | lemma | ring_equiv.to_monoid_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"monoid_hom.id",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom_refl : (ring_equiv.refl R).to_add_monoid_hom = add_monoid_hom.id R | rfl | lemma | ring_equiv.to_add_monoid_hom_refl | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_apply_symm_to_ring_hom_apply (e : R ≃+* S) :
∀ (y : S), e.to_ring_hom (e.symm.to_ring_hom y) = y | e.to_equiv.apply_symm_apply | lemma | ring_equiv.to_ring_hom_apply_symm_to_ring_hom_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_ring_hom_apply_to_ring_hom_apply (e : R ≃+* S) :
∀ (x : R), e.symm.to_ring_hom (e.to_ring_hom x) = x | equiv.symm_apply_apply (e.to_equiv) | lemma | ring_equiv.symm_to_ring_hom_apply_to_ring_hom_apply | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
(e₁.trans e₂).to_ring_hom = e₂.to_ring_hom.comp e₁.to_ring_hom | rfl | lemma | ring_equiv.to_ring_hom_trans | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_comp_symm_to_ring_hom (e : R ≃+* S) :
e.to_ring_hom.comp e.symm.to_ring_hom = ring_hom.id _ | by { ext, simp } | lemma | ring_equiv.to_ring_hom_comp_symm_to_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_ring_hom_comp_to_ring_hom (e : R ≃+* S) :
e.symm.to_ring_hom.comp e.to_ring_hom = ring_hom.id _ | by { ext, simp } | lemma | ring_equiv.symm_to_ring_hom_comp_to_ring_hom | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_hom_inv' {R S F G : Type*} [non_unital_non_assoc_semiring R]
[non_unital_non_assoc_semiring S] [non_unital_ring_hom_class F R S]
[non_unital_ring_hom_class G S R] (hom : F) (inv : G)
(hom_inv_id : (inv : S →ₙ+* R).comp (hom : R →ₙ+* S) = non_unital_ring_hom.id R)
(inv_hom_id : (hom : R →ₙ+* S).comp (inv : S ... | { to_fun := hom,
inv_fun := inv,
left_inv := fun_like.congr_fun hom_inv_id,
right_inv := fun_like.congr_fun inv_hom_id,
map_mul' := map_mul hom,
map_add' := map_add hom, } | def | ring_equiv.of_hom_inv' | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.congr_fun",
"inv_fun",
"map_mul",
"non_unital_non_assoc_semiring",
"non_unital_ring_hom.id",
"non_unital_ring_hom_class"
] | Construct an equivalence of rings from homomorphisms in both directions, which are inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_inv {R S F G : Type*} [non_assoc_semiring R] [non_assoc_semiring S]
[ring_hom_class F R S] [ring_hom_class G S R] (hom : F) (inv : G)
(hom_inv_id : (inv : S →+* R).comp (hom : R →+* S) = ring_hom.id R)
(inv_hom_id : (hom : R →+* S).comp (inv : S →+* R) = ring_hom.id S) :
R ≃+* S | { to_fun := hom,
inv_fun := inv,
left_inv := fun_like.congr_fun hom_inv_id,
right_inv := fun_like.congr_fun inv_hom_id,
map_mul' := map_mul hom,
map_add' := map_add hom, } | def | ring_equiv.of_hom_inv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"fun_like.congr_fun",
"inv_fun",
"map_mul",
"non_assoc_semiring",
"ring_hom.id",
"ring_hom_class"
] | Construct an equivalence of rings from unital homomorphisms in both directions, which are inverses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pow (f : R ≃+* S) (a) : ∀ n : ℕ, f (a ^ n) = (f a) ^ n | map_pow f a | lemma | ring_equiv.map_pow | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_equiv {R S F : Type*} [has_add R] [has_add S] [has_mul R] [has_mul S]
[mul_equiv_class F R S] (f : F) (H : ∀ x y : R, f (x + y) = f x + f y) : R ≃+* S | { ..(f : R ≃* S).to_equiv, ..(f : R ≃* S), ..add_equiv.mk' (f : R ≃* S).to_equiv H } | def | mul_equiv.to_ring_equiv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"mul_equiv_class"
] | Gives a `ring_equiv` from an element of a `mul_equiv_class` preserving addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ring_equiv {R S F : Type*} [has_add R] [has_add S] [has_mul R] [has_mul S]
[add_equiv_class F R S] (f : F) (H : ∀ x y : R, f (x * y) = f x * f y) : R ≃+* S | { ..(f : R ≃+ S).to_equiv, ..(f : R ≃+ S), ..mul_equiv.mk' (f : R ≃+ S).to_equiv H } | def | add_equiv.to_ring_equiv | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"add_equiv_class",
"mul_equiv.mk'"
] | Gives a `ring_equiv` from an element of an `add_equiv_class` preserving addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_trans_symm (e : R ≃+* S) : e.trans e.symm = ring_equiv.refl R | ext e.3 | theorem | ring_equiv.self_trans_symm | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (e : R ≃+* S) : e.symm.trans e = ring_equiv.refl S | ext e.4 | theorem | ring_equiv.symm_trans_self | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_divisors
{A : Type*} (B : Type*) [ring A] [ring B] [no_zero_divisors B]
(e : A ≃+* B) : no_zero_divisors A | { eq_zero_or_eq_zero_of_mul_eq_zero := λ x y hxy,
have e x * e y = 0, by rw [← e.map_mul, hxy, e.map_zero],
by simpa using eq_zero_or_eq_zero_of_mul_eq_zero this } | lemma | ring_equiv.no_zero_divisors | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"no_zero_divisors",
"ring"
] | If two rings are isomorphic, and the second doesn't have zero divisors,
then so does the first. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_domain
{A : Type*} (B : Type*) [ring A] [ring B] [is_domain B]
(e : A ≃+* B) : is_domain A | begin
haveI : nontrivial A := ⟨⟨e.symm 0, e.symm 1, e.symm.injective.ne zero_ne_one⟩⟩,
haveI := e.no_zero_divisors B,
exact no_zero_divisors.to_is_domain _
end | lemma | ring_equiv.is_domain | algebra.ring | src/algebra/ring/equiv.lean | [
"algebra.group.opposite",
"algebra.hom.ring",
"logic.equiv.set",
"tactic.assert_exists"
] | [
"is_domain",
"no_zero_divisors.to_is_domain",
"nontrivial",
"ring"
] | If two rings are isomorphic, and the second is a domain, then so is the first. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_equiv.pi_fin_two (R : fin 2 → Type*) [Π i, semiring (R i)] :
(Π (i : fin 2), R i) ≃+* R 0 × R 1 | { to_fun := pi_fin_two_equiv R,
map_add' := λ a b, rfl,
map_mul' := λ a b, rfl,
.. pi_fin_two_equiv R } | def | ring_equiv.pi_fin_two | algebra.ring | src/algebra/ring/fin.lean | [
"logic.equiv.fin",
"algebra.ring.equiv",
"algebra.group.prod"
] | [
"pi_fin_two_equiv",
"semiring"
] | The product over `fin 2` of some rings is just the cartesian product of these rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_idempotent_elem (p : M) : Prop | p * p = p | def | is_idempotent_elem | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [] | An element `p` is said to be idempotent if `p * p = p` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_is_idempotent [is_idempotent M (*)] (a : M) : is_idempotent_elem a | is_idempotent.idempotent a | lemma | is_idempotent_elem.of_is_idempotent | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq {p : M} (h : is_idempotent_elem p) : p * p = p | h | lemma | is_idempotent_elem.eq | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_of_commute {p q : S} (h : commute p q) (h₁ : is_idempotent_elem p)
(h₂ : is_idempotent_elem q) : is_idempotent_elem (p * q) | by rw [is_idempotent_elem, mul_assoc, ← mul_assoc q, ← h.eq, mul_assoc p, h₂.eq, ← mul_assoc, h₁.eq] | lemma | is_idempotent_elem.mul_of_commute | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"commute",
"is_idempotent_elem",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : is_idempotent_elem (0 : M₀) | mul_zero _ | lemma | is_idempotent_elem.zero | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one : is_idempotent_elem (1 : M₁) | mul_one _ | lemma | is_idempotent_elem.one | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub {p : R} (h : is_idempotent_elem p) : is_idempotent_elem (1 - p) | by rw [is_idempotent_elem, mul_sub, mul_one, sub_mul, one_mul, h.eq, sub_self, sub_zero] | lemma | is_idempotent_elem.one_sub | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_sub_iff {p : R} : is_idempotent_elem (1 - p) ↔ is_idempotent_elem p | ⟨ λ h, sub_sub_cancel 1 p ▸ h.one_sub, is_idempotent_elem.one_sub ⟩ | lemma | is_idempotent_elem.one_sub_iff | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"is_idempotent_elem.one_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow {p : N} (n : ℕ) (h : is_idempotent_elem p) : is_idempotent_elem (p ^ n) | nat.rec_on n ((pow_zero p).symm ▸ one) (λ n ih, show p ^ n.succ * p ^ n.succ = p ^ n.succ,
by { nth_rewrite 2 ←h.eq, rw [←sq, ←sq, ←pow_mul, ←pow_mul'] }) | lemma | is_idempotent_elem.pow | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"ih",
"is_idempotent_elem",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_succ_eq {p : N} (n : ℕ) (h : is_idempotent_elem p) : p ^ (n + 1) = p | nat.rec_on n ((nat.zero_add 1).symm ▸ pow_one p) (λ n ih, by rw [pow_succ, ih, h.eq]) | lemma | is_idempotent_elem.pow_succ_eq | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"ih",
"is_idempotent_elem",
"pow_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_eq_one {p : G} : is_idempotent_elem p ↔ p = 1 | iff.intro (λ h, mul_left_cancel ((mul_one p).symm ▸ h.eq : p * p = p * 1)) (λ h, h.symm ▸ one) | lemma | is_idempotent_elem.iff_eq_one | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"mul_left_cancel",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_eq_zero_or_one {p : G₀} : is_idempotent_elem p ↔ p = 0 ∨ p = 1 | begin
refine iff.intro
(λ h, or_iff_not_imp_left.mpr (λ hp, _))
(λ h, h.elim (λ hp, hp.symm ▸ zero) (λ hp, hp.symm ▸ one)),
exact mul_left_cancel₀ hp (h.trans (mul_one p).symm)
end | lemma | is_idempotent_elem.iff_eq_zero_or_one | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"mul_left_cancel₀",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ↑(0 : {p : M₀ // is_idempotent_elem p}) = (0 : M₀) | rfl | lemma | is_idempotent_elem.coe_zero | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ↑(1 : { p : M₁ // is_idempotent_elem p }) = (1 : M₁) | rfl | lemma | is_idempotent_elem.coe_one | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_compl (p : { p : R // is_idempotent_elem p }) : ↑(pᶜ) = (1 : R) - ↑p | rfl | lemma | is_idempotent_elem.coe_compl | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_compl (p : {p : R // is_idempotent_elem p}) : pᶜᶜ = p | subtype.ext $ sub_sub_cancel _ _ | lemma | is_idempotent_elem.compl_compl | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"compl_compl",
"is_idempotent_elem",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_compl : (0 : {p : R // is_idempotent_elem p})ᶜ = 1 | subtype.ext $ sub_zero _ | lemma | is_idempotent_elem.zero_compl | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_compl : (1 : {p : R // is_idempotent_elem p})ᶜ = 0 | subtype.ext $ sub_self _ | lemma | is_idempotent_elem.one_compl | algebra.ring | src/algebra/ring/idempotents.lean | [
"order.basic",
"algebra.group_power.basic",
"algebra.ring.defs",
"tactic.nth_rewrite"
] | [
"is_idempotent_elem",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.distrib {S} [has_mul R] [has_add R] [distrib S]
(f : R → S) (hf : injective f) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) :
distrib R | { mul := (*),
add := (+),
left_distrib := λ x y z, hf $ by simp only [*, left_distrib],
right_distrib := λ x y z, hf $ by simp only [*, right_distrib] } | def | function.injective.distrib | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"distrib",
"left_distrib",
"right_distrib"
] | Pullback a `distrib` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.distrib {S} [distrib R] [has_add S] [has_mul S]
(f : R → S) (hf : surjective f) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) :
distrib S | { mul := (*),
add := (+),
left_distrib := hf.forall₃.2 $ λ x y z, by simp only [← add, ← mul, left_distrib],
right_distrib := hf.forall₃.2 $ λ x y z, by simp only [← add, ← mul, right_distrib] } | def | function.surjective.distrib | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"distrib",
"left_distrib",
"right_distrib"
] | Pushforward a `distrib` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_unital_non_assoc_semiring
{α : Type u} [non_unital_non_assoc_semiring α]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) :
non_unital_non_assoc_semiring β | { .. hf.mul_zero_class f zero mul, .. hf.add_comm_monoid f zero add nsmul, .. hf.distrib f add mul } | def | function.injective.non_unital_non_assoc_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"non_unital_non_assoc_semiring"
] | Pullback a `non_unital_non_assoc_semiring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_unital_semiring
{α : Type u} [non_unital_semiring α]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) :
non_unital_semiring β | { .. hf.non_unital_non_assoc_semiring f zero add mul nsmul, .. hf.semigroup_with_zero f zero mul } | def | function.injective.non_unital_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"non_unital_semiring"
] | Pullback a `non_unital_semiring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_assoc_semiring
{α : Type u} [non_assoc_semiring α]
{β : Type v} [has_zero β] [has_one β] [has_mul β] [has_add β]
[has_smul ℕ β] [has_nat_cast β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(n... | { .. hf.add_monoid_with_one f zero one add nsmul nat_cast,
.. hf.non_unital_non_assoc_semiring f zero add mul nsmul,
.. hf.mul_one_class f one mul } | def | function.injective.non_assoc_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"non_assoc_semiring"
] | Pullback a `non_assoc_semiring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.semiring
{α : Type u} [semiring α]
{β : Type v} [has_zero β] [has_one β] [has_add β] [has_mul β] [has_pow β ℕ]
[has_smul ℕ β] [has_nat_cast β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul :... | { .. hf.non_assoc_semiring f zero one add mul nsmul nat_cast,
.. hf.monoid_with_zero f zero one mul npow,
.. hf.distrib f add mul } | def | function.injective.semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"semiring"
] | Pullback a `semiring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_unital_non_assoc_semiring
{α : Type u} [non_unital_non_assoc_semiring α]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) :
non_unital_non_assoc_semiring β | { .. hf.mul_zero_class f zero mul, .. hf.add_comm_monoid f zero add nsmul, .. hf.distrib f add mul } | def | function.surjective.non_unital_non_assoc_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"non_unital_non_assoc_semiring"
] | Pushforward a `non_unital_non_assoc_semiring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_unital_semiring
{α : Type u} [non_unital_semiring α]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) :
non_unital_semiring β | { .. hf.non_unital_non_assoc_semiring f zero add mul nsmul, .. hf.semigroup_with_zero f zero mul } | def | function.surjective.non_unital_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"non_unital_semiring"
] | Pushforward a `non_unital_semiring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_assoc_semiring
{α : Type u} [non_assoc_semiring α]
{β : Type v} [has_zero β] [has_one β] [has_add β] [has_mul β]
[has_smul ℕ β] [has_nat_cast β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
... | { .. hf.add_monoid_with_one f zero one add nsmul nat_cast,
.. hf.non_unital_non_assoc_semiring f zero add mul nsmul, .. hf.mul_one_class f one mul } | def | function.surjective.non_assoc_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"non_assoc_semiring"
] | Pushforward a `non_assoc_semiring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.semiring
{α : Type u} [semiring α]
{β : Type v} [has_zero β] [has_one β] [has_add β] [has_mul β] [has_pow β ℕ]
[has_smul ℕ β] [has_nat_cast β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul... | { .. hf.non_assoc_semiring f zero one add mul nsmul nat_cast,
.. hf.monoid_with_zero f zero one mul npow, .. hf.add_comm_monoid f zero add nsmul,
.. hf.distrib f add mul } | def | function.surjective.semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_nat_cast",
"has_smul",
"semiring"
] | Pushforward a `semiring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_unital_comm_semiring [has_zero γ] [has_add γ] [has_mul γ]
[has_smul ℕ γ] (f : γ → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) :
non_unital_comm_semiring γ | { .. hf.non_unital_semiring f zero add mul nsmul, .. hf.comm_semigroup f mul } | def | function.injective.non_unital_comm_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_comm_semiring"
] | Pullback a `non_unital_semiring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_unital_comm_semiring [has_zero γ] [has_add γ] [has_mul γ]
[has_smul ℕ γ] (f : α → γ) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) :
non_unital_comm_semiring γ | { .. hf.non_unital_semiring f zero add mul nsmul, .. hf.comm_semigroup f mul } | def | function.surjective.non_unital_comm_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_comm_semiring"
] | Pushforward a `non_unital_semiring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.comm_semiring
[has_zero γ] [has_one γ] [has_add γ] [has_mul γ] [has_smul ℕ γ] [has_nat_cast γ]
[has_pow γ ℕ] (f : γ → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) (np... | { .. hf.semiring f zero one add mul nsmul npow nat_cast, .. hf.comm_semigroup f mul } | def | function.injective.comm_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"comm_semiring",
"has_nat_cast",
"has_smul"
] | Pullback a `semiring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.comm_semiring
[has_zero γ] [has_one γ] [has_add γ] [has_mul γ] [has_smul ℕ γ] [has_nat_cast γ]
[has_pow γ ℕ] (f : α → γ) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x) (... | { .. hf.semiring f zero one add mul nsmul npow nat_cast, .. hf.comm_semigroup f mul } | def | function.surjective.comm_semiring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"comm_semiring",
"has_nat_cast",
"has_smul"
] | Pushforward a `semiring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.has_distrib_neg [has_neg β] [has_mul β] (f : β → α)
(hf : injective f) (neg : ∀ a, f (-a) = -f a) (mul : ∀ a b, f (a * b) = f a * f b) :
has_distrib_neg β | { neg_mul := λ x y, hf $ by erw [neg, mul, neg, neg_mul, mul],
mul_neg := λ x y, hf $ by erw [neg, mul, neg, mul_neg, mul],
..hf.has_involutive_neg _ neg, ..‹has_mul β› } | def | function.injective.has_distrib_neg | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_distrib_neg",
"mul_neg",
"neg_mul"
] | A type endowed with `-` and `*` has distributive negation, if it admits an injective map that
preserves `-` and `*` to a type which has distributive negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.has_distrib_neg [has_neg β] [has_mul β] (f : α → β)
(hf : surjective f) (neg : ∀ a, f (-a) = -f a) (mul : ∀ a b, f (a * b) = f a * f b) :
has_distrib_neg β | { neg_mul := hf.forall₂.2 $ λ x y, by { erw [←neg, ← mul, neg_mul, neg, mul], refl },
mul_neg := hf.forall₂.2 $ λ x y, by { erw [←neg, ← mul, mul_neg, neg, mul], refl },
..hf.has_involutive_neg _ neg, ..‹has_mul β› } | def | function.surjective.has_distrib_neg | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_distrib_neg",
"mul_neg",
"neg_mul"
] | A type endowed with `-` and `*` has distributive negation, if it admits a surjective map that
preserves `-` and `*` from a type which has distributive negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_unital_non_assoc_ring
[has_zero β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f... | { .. hf.add_comm_group f zero add neg sub nsmul zsmul, ..hf.mul_zero_class f zero mul,
.. hf.distrib f add mul } | def | function.injective.non_unital_non_assoc_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_non_assoc_ring"
] | Pullback a `non_unital_non_assoc_ring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_unital_non_assoc_ring
[has_zero β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) =... | { .. hf.add_comm_group f zero add neg sub nsmul zsmul, .. hf.mul_zero_class f zero mul,
.. hf.distrib f add mul } | def | function.surjective.non_unital_non_assoc_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_non_assoc_ring"
] | Pushforward a `non_unital_non_assoc_ring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_unital_ring
[has_zero β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y)
... | { .. hf.add_comm_group f zero add neg sub nsmul gsmul, ..hf.mul_zero_class f zero mul,
.. hf.distrib f add mul, .. hf.semigroup f mul } | def | function.injective.non_unital_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_ring"
] | Pullback a `non_unital_ring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_unital_ring
[has_zero β] [has_add β] [has_mul β] [has_neg β] [has_sub β] [has_smul ℕ β] [has_smul ℤ β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y... | { .. hf.add_comm_group f zero add neg sub nsmul gsmul, .. hf.mul_zero_class f zero mul,
.. hf.distrib f add mul, .. hf.semigroup f mul } | def | function.surjective.non_unital_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_ring"
] | Pushforward a `non_unital_ring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_assoc_ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β] [has_nat_cast β] [has_int_cast β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(... | { .. hf.add_comm_group f zero add neg sub nsmul gsmul,
.. hf.add_group_with_one f zero one add neg sub nsmul gsmul nat_cast int_cast,
.. hf.mul_zero_class f zero mul, .. hf.distrib f add mul,
.. hf.mul_one_class f one mul } | def | function.injective.non_assoc_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"non_assoc_ring"
] | Pullback a `non_assoc_ring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_assoc_ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β] [has_nat_cast β] [has_int_cast β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
... | { .. hf.add_comm_group f zero add neg sub nsmul gsmul, .. hf.mul_zero_class f zero mul,
.. hf.add_group_with_one f zero one add neg sub nsmul gsmul nat_cast int_cast,
.. hf.distrib f add mul, .. hf.mul_one_class f one mul } | def | function.surjective.non_assoc_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"non_assoc_ring"
] | Pushforward a `non_unital_ring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β] [has_int_cast β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)... | { .. hf.add_group_with_one f zero one add neg sub nsmul zsmul nat_cast int_cast,
.. hf.add_comm_group f zero add neg sub nsmul zsmul,
.. hf.monoid f one mul npow, .. hf.distrib f add mul } | def | function.injective.ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"ring"
] | Pullback a `ring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β] [has_int_cast β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f ... | { .. hf.add_group_with_one f zero one add neg sub nsmul zsmul nat_cast int_cast,
.. hf.add_comm_group f zero add neg sub nsmul zsmul,
.. hf.monoid f one mul npow, .. hf.distrib f add mul } | def | function.surjective.ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_int_cast",
"has_nat_cast",
"has_smul",
"ring"
] | Pushforward a `ring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.non_unital_comm_ring
[has_zero β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β]
(f : β → α) (hf : injective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x ... | { .. hf.non_unital_ring f zero add mul neg sub nsmul zsmul, .. hf.comm_semigroup f mul } | def | function.injective.non_unital_comm_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_comm_ring"
] | Pullback a `comm_ring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.non_unital_comm_ring
[has_zero β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
(neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f ... | { .. hf.non_unital_ring f zero add mul neg sub nsmul zsmul, .. hf.comm_semigroup f mul } | def | function.surjective.non_unital_comm_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul",
"non_unital_comm_ring"
] | Pushforward a `non_unital_comm_ring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.comm_ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β] [has_int_cast β]
(f : β → α) (hf : injective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x *... | { .. hf.ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast,
.. hf.comm_semigroup f mul } | def | function.injective.comm_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"comm_ring",
"has_int_cast",
"has_nat_cast",
"has_smul"
] | Pullback a `comm_ring` instance along an injective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.comm_ring
[has_zero β] [has_one β] [has_add β] [has_mul β] [has_neg β] [has_sub β]
[has_smul ℕ β] [has_smul ℤ β] [has_pow β ℕ] [has_nat_cast β] [has_int_cast β]
(f : α → β) (hf : surjective f) (zero : f 0 = 0) (one : f 1 = 1)
(add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x... | { .. hf.ring f zero one add mul neg sub nsmul zsmul npow nat_cast int_cast,
.. hf.comm_semigroup f mul } | def | function.surjective.comm_ring | algebra.ring | src/algebra/ring/inj_surj.lean | [
"algebra.ring.defs",
"algebra.opposites",
"algebra.group_with_zero.inj_surj"
] | [
"comm_ring",
"has_int_cast",
"has_nat_cast",
"has_smul"
] | Pushforward a `comm_ring` instance along a surjective function.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_ring_hom.to_opposite {R S : Type*} [non_unital_non_assoc_semiring R]
[non_unital_non_assoc_semiring S] (f : R →ₙ+* S) (hf : ∀ x y, commute (f x) (f y)) :
R →ₙ+* Sᵐᵒᵖ | { to_fun := mul_opposite.op ∘ f,
.. ((op_add_equiv : S ≃+ Sᵐᵒᵖ).to_add_monoid_hom.comp ↑f : R →+ Sᵐᵒᵖ),
.. f.to_mul_hom.to_opposite hf } | def | non_unital_ring_hom.to_opposite | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"commute",
"mul_opposite.op",
"non_unital_non_assoc_semiring"
] | A non-unital ring homomorphism `f : R →ₙ+* S` such that `f x` commutes with `f y` for all `x, y`
defines a non-unital ring homomorphism to `Sᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_ring_hom.from_opposite {R S : Type*} [non_unital_non_assoc_semiring R]
[non_unital_non_assoc_semiring S] (f : R →ₙ+* S) (hf : ∀ x y, commute (f x) (f y)) :
Rᵐᵒᵖ →ₙ+* S | { to_fun := f ∘ mul_opposite.unop,
.. (f.to_add_monoid_hom.comp (op_add_equiv : R ≃+ Rᵐᵒᵖ).symm.to_add_monoid_hom : Rᵐᵒᵖ →+ S),
.. f.to_mul_hom.from_opposite hf } | def | non_unital_ring_hom.from_opposite | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"commute",
"mul_opposite.unop",
"non_unital_non_assoc_semiring"
] | A non-unital ring homomorphism `f : R →ₙ* S` such that `f x` commutes with `f y` for all `x, y`
defines a non-unital ring homomorphism from `Rᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_ring_hom.op {α β} [non_unital_non_assoc_semiring α]
[non_unital_non_assoc_semiring β] : (α →ₙ+* β) ≃ (αᵐᵒᵖ →ₙ+* βᵐᵒᵖ) | { to_fun := λ f, { ..f.to_add_monoid_hom.mul_op, ..f.to_mul_hom.op },
inv_fun := λ f, { ..f.to_add_monoid_hom.mul_unop, ..f.to_mul_hom.unop },
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, simp } } | def | non_unital_ring_hom.op | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"inv_fun",
"non_unital_non_assoc_semiring"
] | A non-unital ring hom `α →ₙ+* β` can equivalently be viewed as a non-unital ring hom
`αᵐᵒᵖ →+* βᵐᵒᵖ`. This is the action of the (fully faithful) `ᵐᵒᵖ`-functor on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_ring_hom.unop {α β} [non_unital_non_assoc_semiring α]
[non_unital_non_assoc_semiring β] : (αᵐᵒᵖ →ₙ+* βᵐᵒᵖ) ≃ (α →ₙ+* β) | non_unital_ring_hom.op.symm | def | non_unital_ring_hom.unop | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring"
] | The 'unopposite' of a non-unital ring hom `αᵐᵒᵖ →ₙ+* βᵐᵒᵖ`. Inverse to
`non_unital_ring_hom.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.to_opposite {R S : Type*} [semiring R] [semiring S] (f : R →+* S)
(hf : ∀ x y, commute (f x) (f y)) : R →+* Sᵐᵒᵖ | { to_fun := mul_opposite.op ∘ f,
.. ((op_add_equiv : S ≃+ Sᵐᵒᵖ).to_add_monoid_hom.comp ↑f : R →+ Sᵐᵒᵖ),
.. f.to_monoid_hom.to_opposite hf } | def | ring_hom.to_opposite | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"commute",
"mul_opposite.op",
"semiring"
] | A ring homomorphism `f : R →+* S` such that `f x` commutes with `f y` for all `x, y` defines
a ring homomorphism to `Sᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.from_opposite {R S : Type*} [semiring R] [semiring S] (f : R →+* S)
(hf : ∀ x y, commute (f x) (f y)) : Rᵐᵒᵖ →+* S | { to_fun := f ∘ mul_opposite.unop,
.. (f.to_add_monoid_hom.comp (op_add_equiv : R ≃+ Rᵐᵒᵖ).symm.to_add_monoid_hom : Rᵐᵒᵖ →+ S),
.. f.to_monoid_hom.from_opposite hf } | def | ring_hom.from_opposite | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"commute",
"mul_opposite.unop",
"semiring"
] | A ring homomorphism `f : R →+* S` such that `f x` commutes with `f y` for all `x, y` defines
a ring homomorphism from `Rᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.op {α β} [non_assoc_semiring α] [non_assoc_semiring β] :
(α →+* β) ≃ (αᵐᵒᵖ →+* βᵐᵒᵖ) | { to_fun := λ f, { ..f.to_add_monoid_hom.mul_op, ..f.to_monoid_hom.op },
inv_fun := λ f, { ..f.to_add_monoid_hom.mul_unop, ..f.to_monoid_hom.unop },
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, simp } } | def | ring_hom.op | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"inv_fun",
"non_assoc_semiring"
] | A ring hom `α →+* β` can equivalently be viewed as a ring hom `αᵐᵒᵖ →+* βᵐᵒᵖ`. This is the
action of the (fully faithful) `ᵐᵒᵖ`-functor on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.unop {α β} [non_assoc_semiring α] [non_assoc_semiring β] :
(αᵐᵒᵖ →+* βᵐᵒᵖ) ≃ (α →+* β) | ring_hom.op.symm | def | ring_hom.unop | algebra.ring | src/algebra/ring/opposite.lean | [
"algebra.group_with_zero.basic",
"algebra.group.opposite",
"algebra.hom.ring"
] | [
"non_assoc_semiring"
] | The 'unopposite' of a ring hom `αᵐᵒᵖ →+* βᵐᵒᵖ`. Inverse to `ring_hom.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distrib [Π i, distrib $ f i] : distrib (Π i : I, f i) | by refine_struct { add := (+), mul := (*), .. }; tactic.pi_instance_derive_field | instance | pi.distrib | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"distrib",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_semiring [∀ i, non_unital_non_assoc_semiring $ f i] :
non_unital_non_assoc_semiring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), add := (+), mul := (*), .. };
tactic.pi_instance_derive_field | instance | pi.non_unital_non_assoc_semiring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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