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 |
|---|---|---|---|---|---|---|---|---|---|---|
non_unital_semiring [∀ i, non_unital_semiring $ f i] :
non_unital_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_semiring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_assoc_semiring [∀ i, non_assoc_semiring $ f i] :
non_assoc_semiring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := 1, add := (+), mul := (*), .. };
tactic.pi_instance_derive_field | instance | pi.non_assoc_semiring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring [∀ i, semiring $ f i] : semiring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := 1, add := (+), mul := (*),
nsmul := add_monoid.nsmul, npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.semiring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_comm_semiring [∀ i, non_unital_comm_semiring $ f i] :
non_unital_comm_semiring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), add := (+), mul := (*), nsmul := add_monoid.nsmul };
tactic.pi_instance_derive_field | instance | pi.non_unital_comm_semiring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_comm_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring [∀ i, comm_semiring $ f i] : comm_semiring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := 1, add := (+), mul := (*),
nsmul := add_monoid.nsmul, npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.comm_semiring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"comm_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_ring [∀ i, non_unital_non_assoc_ring $ f i] :
non_unital_non_assoc_ring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), add := (+), mul := (*),
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | pi.non_unital_non_assoc_ring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_ring [∀ i, non_unital_ring $ f i] :
non_unital_ring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), add := (+), mul := (*),
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | pi.non_unital_ring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_assoc_ring [∀ i, non_assoc_ring $ f i] :
non_assoc_ring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), add := (+), mul := (*),
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | pi.non_assoc_ring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring [∀ i, ring $ f i] : ring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := 1, add := (+), mul := (*),
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,
npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.ring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_comm_ring [∀ i, non_unital_comm_ring $ f i] :
non_unital_comm_ring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), add := (+), mul := (*), neg := has_neg.neg,
nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | pi.non_unital_comm_ring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_comm_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring [∀ i, comm_ring $ f i] : comm_ring (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := 1, add := (+), mul := (*),
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,
npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.comm_ring | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"comm_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_ring_hom {γ : Type w} [Π i, non_unital_non_assoc_semiring (f i)]
[non_unital_non_assoc_semiring γ] (g : Π i, γ →ₙ+* f i) : γ →ₙ+* Π i, f i | { to_fun := λ x b, g b x,
.. pi.mul_hom (λ i, (g i).to_mul_hom),
.. pi.add_monoid_hom (λ i, (g i).to_add_monoid_hom) } | def | pi.non_unital_ring_hom | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring",
"non_unital_ring_hom",
"pi.mul_hom"
] | A family of non-unital ring homomorphisms `f a : γ →ₙ+* β a` defines a non-unital ring
homomorphism `pi.non_unital_ring_hom f : γ →+* Π a, β a` given by
`pi.non_unital_ring_hom f x b = f b x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_ring_hom_injective {γ : Type w} [nonempty I]
[Π i, non_unital_non_assoc_semiring (f i)] [non_unital_non_assoc_semiring γ] (g : Π i, γ →ₙ+* f i)
(hg : ∀ i, function.injective (g i)) : function.injective (pi.non_unital_ring_hom g) | mul_hom_injective (λ i, (g i).to_mul_hom) hg | lemma | pi.non_unital_ring_hom_injective | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring",
"pi.non_unital_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom {γ : Type w} [Π i, non_assoc_semiring (f i)] [non_assoc_semiring γ]
(g : Π i, γ →+* f i) : γ →+* Π i, f i | { to_fun := λ x b, g b x,
.. pi.monoid_hom (λ i, (g i).to_monoid_hom),
.. pi.add_monoid_hom (λ i, (g i).to_add_monoid_hom) } | def | pi.ring_hom | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_semiring",
"pi.monoid_hom",
"ring_hom"
] | A family of ring homomorphisms `f a : γ →+* β a` defines a ring homomorphism
`pi.ring_hom f : γ →+* Π a, β a` given by `pi.ring_hom f x b = f b x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_injective {γ : Type w} [nonempty I] [Π i, non_assoc_semiring (f i)]
[non_assoc_semiring γ] (g : Π i, γ →+* f i) (hg : ∀ i, function.injective (g i)) :
function.injective (pi.ring_hom g) | monoid_hom_injective (λ i, (g i).to_monoid_hom) hg | lemma | pi.ring_hom_injective | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_semiring",
"pi.ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.eval_non_unital_ring_hom (f : I → Type v)
[Π i, non_unital_non_assoc_semiring (f i)] (i : I) : (Π i, f i) →ₙ+* f i | { ..(pi.eval_mul_hom f i),
..(pi.eval_add_monoid_hom f i) } | def | pi.eval_non_unital_ring_hom | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring",
"pi.eval_mul_hom"
] | Evaluation of functions into an indexed collection of non-unital rings at a point is a
non-unital ring homomorphism. This is `function.eval` as a `non_unital_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.const_non_unital_ring_hom (α β : Type*) [non_unital_non_assoc_semiring β] : β →ₙ+* (α → β) | { to_fun := function.const _,
.. pi.non_unital_ring_hom (λ _, non_unital_ring_hom.id β) } | def | pi.const_non_unital_ring_hom | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring",
"non_unital_ring_hom.id",
"pi.non_unital_ring_hom"
] | `function.const` as a `non_unital_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_ring_hom.comp_left {α β : Type*} [non_unital_non_assoc_semiring α]
[non_unital_non_assoc_semiring β] (f : α →ₙ+* β) (I : Type*) :
(I → α) →ₙ+* (I → β) | { to_fun := λ h, f ∘ h,
.. f.to_mul_hom.comp_left I,
.. f.to_add_monoid_hom.comp_left I } | def | non_unital_ring_hom.comp_left | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_unital_non_assoc_semiring"
] | Non-unital ring homomorphism between the function spaces `I → α` and `I → β`, induced by a
non-unital ring homomorphism `f` between `α` and `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.eval_ring_hom (f : I → Type v) [Π i, non_assoc_semiring (f i)] (i : I) :
(Π i, f i) →+* f i | { ..(pi.eval_monoid_hom f i),
..(pi.eval_add_monoid_hom f i) } | def | pi.eval_ring_hom | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_semiring",
"pi.eval_monoid_hom"
] | Evaluation of functions into an indexed collection of rings at a point is a ring
homomorphism. This is `function.eval` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.const_ring_hom (α β : Type*) [non_assoc_semiring β] : β →+* (α → β) | { to_fun := function.const _,
.. pi.ring_hom (λ _, ring_hom.id β) } | def | pi.const_ring_hom | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_semiring",
"pi.ring_hom",
"ring_hom.id"
] | `function.const` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.comp_left {α β : Type*} [non_assoc_semiring α]
[non_assoc_semiring β] (f : α →+* β) (I : Type*) :
(I → α) →+* (I → β) | { to_fun := λ h, f ∘ h,
.. f.to_monoid_hom.comp_left I,
.. f.to_add_monoid_hom.comp_left I } | def | ring_hom.comp_left | algebra.ring | src/algebra/ring/pi.lean | [
"tactic.pi_instances",
"algebra.group.pi",
"algebra.hom.ring"
] | [
"non_assoc_semiring"
] | Ring homomorphism between the function spaces `I → α` and `I → β`, induced by a ring
homomorphism `f` between `α` and `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst : R × S →ₙ+* R | { to_fun := prod.fst, .. mul_hom.fst R S, .. add_monoid_hom.fst R S } | def | non_unital_ring_hom.fst | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"mul_hom.fst"
] | Given non-unital semirings `R`, `S`, the natural projection homomorphism from `R × S` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : R × S →ₙ+* S | { to_fun := prod.snd, .. mul_hom.snd R S, .. add_monoid_hom.snd R S } | def | non_unital_ring_hom.snd | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"mul_hom.snd"
] | Given non-unital semirings `R`, `S`, the natural projection homomorphism from `R × S` to `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fst : ⇑(fst R S) = prod.fst | rfl | lemma | non_unital_ring_hom.coe_fst | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_snd : ⇑(snd R S) = prod.snd | rfl | lemma | non_unital_ring_hom.coe_snd | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (f : R →ₙ+* S) (g : R →ₙ+* T) : R →ₙ+* S × T | { to_fun := λ x, (f x, g x),
.. mul_hom.prod (f : mul_hom R S) (g : mul_hom R T),
.. add_monoid_hom.prod (f : R →+ S) (g : R →+ T) } | def | non_unital_ring_hom.prod | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"mul_hom",
"mul_hom.prod"
] | Combine two non-unital ring homomorphisms `f : R →ₙ+* S`, `g : R →ₙ+* T` into
`f.prod g : R →ₙ+* S × T` given by `(f.prod g) x = (f x, g x)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_apply (x) : f.prod g x = (f x, g x) | rfl | lemma | non_unital_ring_hom.prod_apply | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_prod : (fst S T).comp (f.prod g) = f | ext $ λ x, rfl | lemma | non_unital_ring_hom.fst_comp_prod | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_prod : (snd S T).comp (f.prod g) = g | ext $ λ x, rfl | lemma | non_unital_ring_hom.snd_comp_prod | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_unique (f : R →ₙ+* S × T) :
((fst S T).comp f).prod ((snd S T).comp f) = f | ext $ λ x, by simp only [prod_apply, coe_fst, coe_snd, comp_apply, prod.mk.eta] | lemma | non_unital_ring_hom.prod_unique | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map : R × S →ₙ+* R' × S' | (f.comp (fst R S)).prod (g.comp (snd R S)) | def | non_unital_ring_hom.prod_map | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"prod_map"
] | `prod.map` as a `non_unital_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map_def : prod_map f g = (f.comp (fst R S)).prod (g.comp (snd R S)) | rfl | lemma | non_unital_ring_hom.prod_map_def | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comp_prod_map (f : T →ₙ+* R) (g : T →ₙ+* S) (f' : R →ₙ+* R') (g' : S →ₙ+* S') :
(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g) | rfl | lemma | non_unital_ring_hom.prod_comp_prod_map | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : R × S →+* R | { to_fun := prod.fst, .. monoid_hom.fst R S, .. add_monoid_hom.fst R S } | def | ring_hom.fst | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"monoid_hom.fst"
] | Given semirings `R`, `S`, the natural projection homomorphism from `R × S` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : R × S →+* S | { to_fun := prod.snd, .. monoid_hom.snd R S, .. add_monoid_hom.snd R S } | def | ring_hom.snd | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"monoid_hom.snd"
] | Given semirings `R`, `S`, the natural projection homomorphism from `R × S` to `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : R →+* S) (g : R →+* T) : R →+* S × T | { to_fun := λ x, (f x, g x),
.. monoid_hom.prod (f : R →* S) (g : R →* T), .. add_monoid_hom.prod (f : R →+ S) (g : R →+ T) } | def | ring_hom.prod | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"monoid_hom.prod"
] | Combine two ring homomorphisms `f : R →+* S`, `g : R →+* T` into `f.prod g : R →+* S × T`
given by `(f.prod g) x = (f x, g x)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_unique (f : R →+* S × T) :
((fst S T).comp f).prod ((snd S T).comp f) = f | ext $ λ x, by simp only [prod_apply, coe_fst, coe_snd, comp_apply, prod.mk.eta] | lemma | ring_hom.prod_unique | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map : R × S →+* R' × S' | (f.comp (fst R S)).prod (g.comp (snd R S)) | def | ring_hom.prod_map | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"prod_map"
] | `prod.map` as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comp_prod_map (f : T →+* R) (g : T →+* S) (f' : R →+* R') (g' : S →+* S') :
(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g) | rfl | lemma | ring_hom.prod_comp_prod_map | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm : R × S ≃+* S × R | { ..add_equiv.prod_comm, ..mul_equiv.prod_comm } | def | ring_equiv.prod_comm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"mul_equiv.prod_comm"
] | Swapping components as an equivalence of (semi)rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_comm : ⇑(prod_comm : R × S ≃+* S × R) = prod.swap | rfl | lemma | ring_equiv.coe_prod_comm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_comm_symm : ⇑((prod_comm : R × S ≃+* S × R).symm) = prod.swap | rfl | lemma | ring_equiv.coe_prod_comm_symm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_coe_prod_comm :
(ring_hom.fst S R).comp ↑(prod_comm : R × S ≃+* S × R) = ring_hom.snd R S | ring_hom.ext $ λ _, rfl | lemma | ring_equiv.fst_comp_coe_prod_comm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"ring_hom.ext",
"ring_hom.fst",
"ring_hom.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_coe_prod_comm :
(ring_hom.snd S R).comp ↑(prod_comm : R × S ≃+* S × R) = ring_hom.fst R S | ring_hom.ext $ λ _, rfl | lemma | ring_equiv.snd_comp_coe_prod_comm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"ring_hom.ext",
"ring_hom.fst",
"ring_hom.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_prod_prod_comm : (R × R') × (S × S') ≃+* (R × S) × (R' × S') | { to_fun := λ rrss, ((rrss.1.1, rrss.2.1), (rrss.1.2, rrss.2.2)),
inv_fun := λ rsrs, ((rsrs.1.1, rsrs.2.1), (rsrs.1.2, rsrs.2.2)),
.. add_equiv.prod_prod_prod_comm R R' S S',
.. mul_equiv.prod_prod_prod_comm R R' S S' } | def | ring_equiv.prod_prod_prod_comm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"inv_fun",
"mul_equiv.prod_prod_prod_comm"
] | Four-way commutativity of `prod`. The name matches `mul_mul_mul_comm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_prod_prod_comm_symm :
(prod_prod_prod_comm R R' S S').symm = prod_prod_prod_comm R S R' S' | rfl | lemma | ring_equiv.prod_prod_prod_comm_symm | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_prod_prod_comm_to_add_equiv :
(prod_prod_prod_comm R R' S S').to_add_equiv = add_equiv.prod_prod_prod_comm R R' S S' | rfl | lemma | ring_equiv.prod_prod_prod_comm_to_add_equiv | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_prod_prod_comm_to_mul_equiv :
(prod_prod_prod_comm R R' S S').to_mul_equiv = mul_equiv.prod_prod_prod_comm R R' S S' | rfl | lemma | ring_equiv.prod_prod_prod_comm_to_mul_equiv | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"mul_equiv.prod_prod_prod_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_prod_prod_comm_to_equiv :
(prod_prod_prod_comm R R' S S').to_equiv = equiv.prod_prod_prod_comm R R' S S' | rfl | lemma | ring_equiv.prod_prod_prod_comm_to_equiv | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"equiv.prod_prod_prod_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_zero_ring : R ≃+* R × S | { to_fun := λ x, (x, 0),
inv_fun := prod.fst,
map_add' := by simp,
map_mul' := by simp,
left_inv := λ x, rfl,
right_inv := λ x, by cases x; simp } | def | ring_equiv.prod_zero_ring | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"inv_fun"
] | A ring `R` is isomorphic to `R × S` when `S` is the zero ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_ring_prod : R ≃+* S × R | { to_fun := λ x, (0, x),
inv_fun := prod.snd,
map_add' := by simp,
map_mul' := by simp,
left_inv := λ x, rfl,
right_inv := λ x, by cases x; simp } | def | ring_equiv.zero_ring_prod | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"inv_fun"
] | A ring `R` is isomorphic to `S × R` when `S` is the zero ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
false_of_nontrivial_of_product_domain (R S : Type*) [ring R] [ring S]
[is_domain (R × S)] [nontrivial R] [nontrivial S] : false | begin
have := no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero
(show ((0 : R), (1 : S)) * (1, 0) = 0, by simp),
rw [prod.mk_eq_zero,prod.mk_eq_zero] at this,
rcases this with (⟨_,h⟩|⟨h,_⟩),
{ exact zero_ne_one h.symm },
{ exact zero_ne_one h.symm }
end | lemma | false_of_nontrivial_of_product_domain | algebra.ring | src/algebra/ring/prod.lean | [
"data.int.cast.prod",
"algebra.group.prod",
"algebra.ring.equiv",
"algebra.order.monoid.prod"
] | [
"is_domain",
"nontrivial",
"ring",
"zero_ne_one"
] | The product of two nontrivial rings is not a domain | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular_of_non_zero_divisor [non_unital_non_assoc_ring α] (k : α)
(h : ∀ (x : α), k * x = 0 → x = 0) : is_left_regular k | begin
refine λ x y (h' : k * x = k * y), sub_eq_zero.mp (h _ _),
rw [mul_sub, sub_eq_zero, h']
end | lemma | is_left_regular_of_non_zero_divisor | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"is_left_regular",
"non_unital_non_assoc_ring"
] | Left `mul` by a `k : α` over `[ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `no_zero_divisors`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular_of_non_zero_divisor [non_unital_non_assoc_ring α] (k : α)
(h : ∀ (x : α), x * k = 0 → x = 0) : is_right_regular k | begin
refine λ x y (h' : x * k = y * k), sub_eq_zero.mp (h _ _),
rw [sub_mul, sub_eq_zero, h']
end | lemma | is_right_regular_of_non_zero_divisor | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"is_right_regular",
"non_unital_non_assoc_ring"
] | Right `mul` by a `k : α` over `[ring α]` is injective, if `k` is not a zero divisor.
The typeclass that restricts all terms of `α` to have this property is `no_zero_divisors`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_of_ne_zero' [non_unital_non_assoc_ring α] [no_zero_divisors α] {k : α}
(hk : k ≠ 0) : is_regular k | ⟨is_left_regular_of_non_zero_divisor k
(λ x h, (no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_left hk),
is_right_regular_of_non_zero_divisor k
(λ x h, (no_zero_divisors.eq_zero_or_eq_zero_of_mul_eq_zero h).resolve_right hk)⟩ | lemma | is_regular_of_ne_zero' | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"is_regular",
"is_right_regular_of_non_zero_divisor",
"no_zero_divisors",
"non_unital_non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_iff_ne_zero' [nontrivial α] [non_unital_non_assoc_ring α] [no_zero_divisors α]
{k : α} : is_regular k ↔ k ≠ 0 | ⟨λ h, by { rintro rfl, exact not_not.mpr h.left not_is_left_regular_zero }, is_regular_of_ne_zero'⟩ | lemma | is_regular_iff_ne_zero' | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"is_regular",
"no_zero_divisors",
"non_unital_non_assoc_ring",
"nontrivial",
"not_is_left_regular_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_divisors.to_cancel_monoid_with_zero [ring α] [no_zero_divisors α] :
cancel_monoid_with_zero α | { mul_left_cancel_of_ne_zero := λ a b c ha,
@is_regular.left _ _ _ (is_regular_of_ne_zero' ha) _ _,
mul_right_cancel_of_ne_zero := λ a b c hb,
@is_regular.right _ _ _ (is_regular_of_ne_zero' hb) _ _,
.. (by apply_instance : monoid_with_zero α) } | def | no_zero_divisors.to_cancel_monoid_with_zero | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"cancel_monoid_with_zero",
"is_regular_of_ne_zero'",
"monoid_with_zero",
"no_zero_divisors",
"ring"
] | A ring with no zero divisors is a `cancel_monoid_with_zero`.
Note this is not an instance as it forms a typeclass loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_divisors.to_cancel_comm_monoid_with_zero [comm_ring α] [no_zero_divisors α] :
cancel_comm_monoid_with_zero α | { .. no_zero_divisors.to_cancel_monoid_with_zero,
.. (by apply_instance : comm_monoid_with_zero α) } | def | no_zero_divisors.to_cancel_comm_monoid_with_zero | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"cancel_comm_monoid_with_zero",
"comm_monoid_with_zero",
"comm_ring",
"no_zero_divisors",
"no_zero_divisors.to_cancel_monoid_with_zero"
] | A commutative ring with no zero divisors is a `cancel_comm_monoid_with_zero`.
Note this is not an instance as it forms a typeclass loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_domain.to_cancel_monoid_with_zero [semiring α] [is_domain α] :
cancel_monoid_with_zero α | { .. semiring.to_monoid_with_zero α, .. ‹is_domain α› } | instance | is_domain.to_cancel_monoid_with_zero | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"cancel_monoid_with_zero",
"is_domain",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain.to_cancel_comm_monoid_with_zero : cancel_comm_monoid_with_zero α | { .. ‹comm_semiring α›, .. ‹is_domain α› } | instance | is_domain.to_cancel_comm_monoid_with_zero | algebra.ring | src/algebra/ring/regular.lean | [
"algebra.regular.basic",
"algebra.ring.defs"
] | [
"cancel_comm_monoid_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right [distrib R] {a x y x' y' : R}
(h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x + x') (y + y') | by simp only [semiconj_by, left_distrib, right_distrib, h.eq, h'.eq] | lemma | semiconj_by.add_right | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"distrib",
"left_distrib",
"right_distrib",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left [distrib R] {a b x y : R}
(ha : semiconj_by a x y) (hb : semiconj_by b x y) :
semiconj_by (a + b) x y | by simp only [semiconj_by, left_distrib, right_distrib, ha.eq, hb.eq] | lemma | semiconj_by.add_left | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"distrib",
"left_distrib",
"right_distrib",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_right (h : semiconj_by a x y) : semiconj_by a (-x) (-y) | by simp only [semiconj_by, h.eq, neg_mul, mul_neg] | lemma | semiconj_by.neg_right | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"mul_neg",
"neg_mul",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_right_iff : semiconj_by a (-x) (-y) ↔ semiconj_by a x y | ⟨λ h, neg_neg x ▸ neg_neg y ▸ h.neg_right, semiconj_by.neg_right⟩ | lemma | semiconj_by.neg_right_iff | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_left (h : semiconj_by a x y) : semiconj_by (-a) x y | by simp only [semiconj_by, h.eq, neg_mul, mul_neg] | lemma | semiconj_by.neg_left | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"mul_neg",
"neg_mul",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_left_iff : semiconj_by (-a) x y ↔ semiconj_by a x y | ⟨λ h, neg_neg a ▸ h.neg_left, semiconj_by.neg_left⟩ | lemma | semiconj_by.neg_left_iff | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_right (a : R) : semiconj_by a (-1) (-1) | (one_right a).neg_right | lemma | semiconj_by.neg_one_right | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_left (x : R) : semiconj_by (-1) x x | (semiconj_by.one_left x).neg_left | lemma | semiconj_by.neg_one_left | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"semiconj_by",
"semiconj_by.one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_right (h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x - x') (y - y') | by simpa only [sub_eq_add_neg] using h.add_right h'.neg_right | lemma | semiconj_by.sub_right | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_left (ha : semiconj_by a x y) (hb : semiconj_by b x y) :
semiconj_by (a - b) x y | by simpa only [sub_eq_add_neg] using ha.add_left hb.neg_left | lemma | semiconj_by.sub_left | algebra.ring | src/algebra/ring/semiconj.lean | [
"algebra.group.semiconj",
"algebra.ring.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_class [mul_zero_class α] : mul_zero_class (ulift α) | by refine_struct { zero := (0 : ulift α), mul := (*), .. }; tactic.pi_instance_derive_field | instance | ulift.mul_zero_class | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"mul_zero_class",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib [distrib α] : distrib (ulift α) | by refine_struct { add := (+), mul := (*), .. }; tactic.pi_instance_derive_field | instance | ulift.distrib | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"distrib",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_semiring [non_unital_non_assoc_semiring α] :
non_unital_non_assoc_semiring (ulift α) | by refine_struct { zero := (0 : ulift α), add := (+), mul := (*),
nsmul := add_monoid.nsmul, };
tactic.pi_instance_derive_field | instance | ulift.non_unital_non_assoc_semiring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_unital_non_assoc_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_assoc_semiring [non_assoc_semiring α] : non_assoc_semiring (ulift α) | by refine_struct { zero := (0 : ulift α), one := 1, add := (+), mul := (*),
nsmul := add_monoid.nsmul, .. ulift.add_monoid_with_one };
tactic.pi_instance_derive_field | instance | ulift.non_assoc_semiring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_assoc_semiring",
"tactic.pi_instance_derive_field",
"ulift.add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_semiring [non_unital_semiring α] : non_unital_semiring (ulift α) | by refine_struct { zero := (0 : ulift α), add := (+), mul := (*),
nsmul := add_monoid.nsmul, };
tactic.pi_instance_derive_field | instance | ulift.non_unital_semiring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_unital_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring [semiring α] : semiring (ulift α) | by refine_struct { zero := (0 : ulift α), one := 1, add := (+), mul := (*),
nsmul := add_monoid.nsmul, npow := monoid.npow, .. ulift.add_monoid_with_one };
tactic.pi_instance_derive_field | instance | ulift.semiring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"semiring",
"tactic.pi_instance_derive_field",
"ulift.add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv [non_unital_non_assoc_semiring α] : ulift α ≃+* α | { to_fun := ulift.down,
inv_fun := ulift.up,
map_mul' := λ x y, rfl,
map_add' := λ x y, rfl,
left_inv := by tidy,
right_inv := by tidy, } | def | ulift.ring_equiv | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"inv_fun",
"non_unital_non_assoc_semiring",
"ring_equiv"
] | The ring equivalence between `ulift α` and `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_comm_semiring [non_unital_comm_semiring α] :
non_unital_comm_semiring (ulift α) | by refine_struct { zero := (0 : ulift α), add := (+), mul := (*), nsmul := add_monoid.nsmul };
tactic.pi_instance_derive_field | instance | ulift.non_unital_comm_semiring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_unital_comm_semiring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring [comm_semiring α] : comm_semiring (ulift α) | by refine_struct { zero := (0 : ulift α), one := 1, add := (+), mul := (*),
nsmul := add_monoid.nsmul, npow := monoid.npow, .. ulift.semiring };
tactic.pi_instance_derive_field | instance | ulift.comm_semiring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"comm_semiring",
"tactic.pi_instance_derive_field",
"ulift.semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_ring [non_unital_non_assoc_ring α] :
non_unital_non_assoc_ring (ulift α) | by refine_struct { zero := (0 : ulift α), add := (+), mul := (*), sub := has_sub.sub,
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | ulift.non_unital_non_assoc_ring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_unital_non_assoc_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_ring [non_unital_ring α] :
non_unital_ring (ulift α) | by refine_struct { zero := (0 : ulift α), add := (+), mul := (*), sub := has_sub.sub,
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | ulift.non_unital_ring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_unital_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_assoc_ring [non_assoc_ring α] :
non_assoc_ring (ulift α) | by refine_struct { zero := (0 : ulift α), one := 1, add := (+), mul := (*), sub := has_sub.sub,
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,
.. ulift.add_group_with_one };
tactic.pi_instance_derive_field | instance | ulift.non_assoc_ring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_assoc_ring",
"tactic.pi_instance_derive_field",
"ulift.add_group_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring [ring α] : ring (ulift α) | by refine_struct { zero := (0 : ulift α), one := 1, add := (+), mul := (*), sub := has_sub.sub,
neg := has_neg.neg, nsmul := add_monoid.nsmul, npow := monoid.npow,
zsmul := sub_neg_monoid.zsmul, .. ulift.semiring, .. ulift.add_group_with_one };
tactic.pi_instance_derive_field | instance | ulift.ring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"ring",
"tactic.pi_instance_derive_field",
"ulift.add_group_with_one",
"ulift.semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_comm_ring [non_unital_comm_ring α] : non_unital_comm_ring (ulift α) | by refine_struct { zero := (0 : ulift α), add := (+), mul := (*), sub := has_sub.sub,
neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };
tactic.pi_instance_derive_field | instance | ulift.non_unital_comm_ring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"non_unital_comm_ring",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring [comm_ring α] : comm_ring (ulift α) | by refine_struct { .. ulift.ring };
tactic.pi_instance_derive_field | instance | ulift.comm_ring | algebra.ring | src/algebra/ring/ulift.lean | [
"algebra.group.ulift",
"algebra.ring.equiv"
] | [
"comm_ring",
"tactic.pi_instance_derive_field",
"ulift.ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (u : αˣ) : (↑-u : α) = -u | rfl | theorem | units.coe_neg | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [] | Representing an element of a ring's unit group as an element of the ring commutes with
mapping this element to its additive inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_neg_one : ((-1 : αˣ) : α) = -1 | rfl | theorem | units.coe_neg_one | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_divp (a : α) (u : αˣ) : -(a /ₚ u) = (-a) /ₚ u | by simp only [divp, neg_mul] | lemma | units.neg_divp | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"divp",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_add_divp_same (a b : α) (u : αˣ) :
a /ₚ u + b /ₚ u = (a + b) /ₚ u | by simp only [divp, add_mul] | lemma | units.divp_add_divp_same | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"divp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_sub_divp_same (a b : α) (u : αˣ) :
a /ₚ u - b /ₚ u = (a - b) /ₚ u | by rw [sub_eq_add_neg, sub_eq_add_neg, neg_divp, divp_add_divp_same] | lemma | units.divp_sub_divp_same | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_divp (a b : α) (u : αˣ) : a + b /ₚ u = (a * u + b) /ₚ u | by simp only [divp, add_mul, units.mul_inv_cancel_right] | lemma | units.add_divp | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"divp",
"units.mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_divp (a b : α) (u : αˣ) : a - b /ₚ u = (a * u - b) /ₚ u | by simp only [divp, sub_mul, units.mul_inv_cancel_right] | lemma | units.sub_divp | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"divp",
"units.mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_add (a b : α) (u : αˣ) : a /ₚ u + b = (a + b * u) /ₚ u | by simp only [divp, add_mul, units.mul_inv_cancel_right] | lemma | units.divp_add | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"divp",
"units.mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_sub (a b : α) (u : αˣ) : a /ₚ u - b = (a - b * u) /ₚ u | begin
simp only [divp, sub_mul, sub_right_inj],
assoc_rw [units.mul_inv, mul_one],
end | lemma | units.divp_sub | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"divp",
"mul_one",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.neg [monoid α] [has_distrib_neg α] {a : α} : is_unit a → is_unit (-a) | | ⟨x, hx⟩ := hx ▸ (-x).is_unit | lemma | is_unit.neg | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"has_distrib_neg",
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.neg_iff [monoid α] [has_distrib_neg α] (a : α) : is_unit (-a) ↔ is_unit a | ⟨λ h, neg_neg a ▸ h.neg, is_unit.neg⟩ | lemma | is_unit.neg_iff | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"has_distrib_neg",
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.sub_iff [ring α] {x y : α} :
is_unit (x - y) ↔ is_unit (y - x) | (is_unit.neg_iff _).symm.trans $ neg_sub x y ▸ iff.rfl | lemma | is_unit.sub_iff | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"is_unit",
"is_unit.neg_iff",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_add_divp [comm_ring α] (a b : α) (u₁ u₂ : αˣ) :
a /ₚ u₁ + b /ₚ u₂ = (a * u₂ + u₁ * b) /ₚ (u₁ * u₂) | begin
simp only [divp, add_mul, mul_inv_rev, coe_mul],
rw [mul_comm (↑u₁ * b), mul_comm b],
assoc_rw [mul_inv, mul_inv, mul_one, mul_one],
end | lemma | units.divp_add_divp | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"comm_ring",
"divp",
"mul_comm",
"mul_inv",
"mul_inv_rev",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_sub_divp [comm_ring α] (a b : α) (u₁ u₂ : αˣ) :
(a /ₚ u₁) - (b /ₚ u₂) = ((a * u₂) - (u₁ * b)) /ₚ (u₁ * u₂) | by simp_rw [sub_eq_add_neg, neg_divp, divp_add_divp, mul_neg] | lemma | units.divp_sub_divp | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"comm_ring",
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_mul_one_add_div [semiring R] {a : Rˣ} {b : R} : ↑a + b = a * (1 + ↑a⁻¹ * b) | by rwa [mul_add, mul_one, ← mul_assoc, units.mul_inv, one_mul] | lemma | units.add_eq_mul_one_add_div | algebra.ring | src/algebra/ring/units.lean | [
"algebra.ring.inj_surj",
"algebra.group.units"
] | [
"mul_assoc",
"mul_one",
"one_mul",
"semiring",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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