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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|---|---|---|---|---|---|---|---|---|---|---|
coe_copy (f : α →+* β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | ring_hom.coe_copy | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →+* β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | ring_hom.copy_eq | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x | fun_like.congr_fun h x | lemma | ring_hom.congr_fun | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg (f : α →+* β) {x y : α} (h : x = y) : f x = f y | fun_like.congr_arg f h | lemma | ring_hom.congr_arg | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.congr_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj ⦃f g : α →+* β⦄ (h : (f : α → β) = g) : f = g | fun_like.coe_injective h | lemma | ring_hom.coe_inj | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃f g : α →+* β⦄ : (∀ x, f x = g x) → f = g | fun_like.ext _ _ | lemma | ring_hom.ext | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : α →+* β} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | ring_hom.ext_iff | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : α →+* β) (h₁ h₂ h₃ h₄) : ring_hom.mk f h₁ h₂ h₃ h₄ = f | ext $ λ _, rfl | lemma | ring_hom.mk_coe | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_injective : injective (coe : (α →+* β) → (α →+ β)) | λ f g h, ext $ add_monoid_hom.congr_fun h | lemma | ring_hom.coe_add_monoid_hom_injective | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_injective : injective (coe : (α →+* β) → (α →* β)) | λ f g h, ext $ monoid_hom.congr_fun h | lemma | ring_hom.coe_monoid_hom_injective | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"monoid_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero (f : α →+* β) : f 0 = 0 | map_zero f | lemma | ring_hom.map_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Ring homomorphisms map zero to zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_one (f : α →+* β) : f 1 = 1 | map_one f | lemma | ring_hom.map_one | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_one"
] | Ring homomorphisms map one to one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_add (f : α →+* β) : ∀ a b, f (a + b) = f a + f b | map_add f | lemma | ring_hom.map_add | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Ring homomorphisms preserve addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mul (f : α →+* β) : ∀ a b, f (a * b) = f a * f b | map_mul f | lemma | ring_hom.map_mul | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_mul"
] | Ring homomorphisms preserve multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_bit0 (f : α →+* β) : ∀ a, f (bit0 a) = bit0 (f a) | map_bit0 f | lemma | ring_hom.map_bit0 | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_bit0"
] | Ring homomorphisms preserve `bit0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_bit1 (f : α →+* β) : ∀ a, f (bit1 a) = bit1 (f a) | map_bit1 f | lemma | ring_hom.map_bit1 | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_bit1"
] | Ring homomorphisms preserve `bit1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_ite_zero_one {F : Type*} [ring_hom_class F α β] (f : F) (p : Prop) [decidable p] :
f (ite p 0 1) = ite p 0 1 | by { split_ifs; simp [h] } | lemma | ring_hom.map_ite_zero_one | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ite_one_zero {F : Type*} [ring_hom_class F α β] (f : F) (p : Prop) [decidable p] :
f (ite p 1 0) = ite p 1 0 | by { split_ifs; simp [h] } | lemma | ring_hom.map_ite_one_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codomain_trivial_iff_map_one_eq_zero : (0 : β) = 1 ↔ f 1 = 0 | by rw [map_one, eq_comm] | lemma | ring_hom.codomain_trivial_iff_map_one_eq_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_one"
] | `f : α →+* β` has a trivial codomain iff `f 1 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
codomain_trivial_iff_range_trivial : (0 : β) = 1 ↔ ∀ x, f x = 0 | f.codomain_trivial_iff_map_one_eq_zero.trans
⟨λ h x, by rw [←mul_one x, map_mul, h, mul_zero], λ h, h 1⟩ | lemma | ring_hom.codomain_trivial_iff_range_trivial | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_mul",
"mul_zero"
] | `f : α →+* β` has a trivial codomain iff it has a trivial range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
codomain_trivial_iff_range_eq_singleton_zero : (0 : β) = 1 ↔ set.range f = {0} | f.codomain_trivial_iff_range_trivial.trans
⟨ λ h, set.ext (λ y, ⟨λ ⟨x, hx⟩, by simp [←hx, h x], λ hy, ⟨0, by simpa using hy.symm⟩⟩),
λ h x, set.mem_singleton_iff.mp (h ▸ set.mem_range_self x)⟩ | lemma | ring_hom.codomain_trivial_iff_range_eq_singleton_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"set.ext",
"set.mem_range_self",
"set.range"
] | `f : α →+* β` has a trivial codomain iff its range is `{0}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_one_ne_zero [nontrivial β] : f 1 ≠ 0 | mt f.codomain_trivial_iff_map_one_eq_zero.mpr zero_ne_one | lemma | ring_hom.map_one_ne_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"nontrivial",
"zero_ne_one"
] | `f : α →+* β` doesn't map `1` to `0` if `β` is nontrivial | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
domain_nontrivial [nontrivial β] : nontrivial α | ⟨⟨1, 0, mt (λ h, show f 1 = 0, by rw [h, map_zero]) f.map_one_ne_zero⟩⟩ | lemma | ring_hom.domain_nontrivial | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"nontrivial"
] | If there is a homomorphism `f : α →+* β` and `β` is nontrivial, then `α` is nontrivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
codomain_trivial (f : α →+* β) [h : subsingleton α] : subsingleton β | (subsingleton_or_nontrivial β).resolve_right
(λ _, by exactI not_nontrivial_iff_subsingleton.mpr h f.domain_nontrivial) | lemma | ring_hom.codomain_trivial | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"subsingleton_or_nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg [non_assoc_ring α] [non_assoc_ring β] (f : α →+* β) (x : α) :
f (-x) = -(f x) | map_neg f x | theorem | ring_hom.map_neg | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_assoc_ring"
] | Ring homomorphisms preserve additive inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_sub [non_assoc_ring α] [non_assoc_ring β] (f : α →+* β) (x y : α) :
f (x - y) = (f x) - (f y) | map_sub f x y | theorem | ring_hom.map_sub | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_assoc_ring"
] | Ring homomorphisms preserve subtraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' [non_assoc_semiring α] [non_assoc_ring β] (f : α →* β)
(map_add : ∀ a b, f (a + b) = f a + f b) :
α →+* β | { ..add_monoid_hom.mk' f map_add, ..f } | def | ring_hom.mk' | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"mk'",
"non_assoc_ring",
"non_assoc_semiring"
] | Makes a ring homomorphism from a monoid homomorphism of rings which preserves addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_map (f : α →+* β) {a : α} : is_unit a → is_unit (f a) | is_unit.map f | lemma | ring_hom.is_unit_map | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"is_unit",
"is_unit.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_dvd (f : α →+* β) {a b : α} : a ∣ b → f a ∣ f b | map_dvd f | lemma | ring_hom.map_dvd | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"map_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (α : Type*) [non_assoc_semiring α] : α →+* α | by refine {to_fun := id, ..}; intros; refl | def | ring_hom.id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_assoc_semiring"
] | The identity ring homomorphism from a semiring to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (x : α) : ring_hom.id α x = x | rfl | lemma | ring_hom.id_apply | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_id : (id α : α →+ α) = add_monoid_hom.id α | rfl | lemma | ring_hom.coe_add_monoid_hom_id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_id : (id α : α →* α) = monoid_hom.id α | rfl | lemma | ring_hom.coe_monoid_hom_id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : β →+* γ) (f : α →+* β) : α →+* γ | { to_fun := g ∘ f,
map_one' := by simp,
..g.to_non_unital_ring_hom.comp f.to_non_unital_ring_hom } | def | ring_hom.comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | Composition of ring homomorphisms is a ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_assoc {δ} {rδ: non_assoc_semiring δ} (f : α →+* β) (g : β →+* γ) (h : γ →+* δ) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | ring_hom.comp_assoc | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"non_assoc_semiring"
] | Composition of semiring homomorphisms is associative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (hnp : β →+* γ) (hmn : α →+* β) : (hnp.comp hmn : α → γ) = hnp ∘ hmn | rfl | lemma | ring_hom.coe_comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (hnp : β →+* γ) (hmn : α →+* β) (x : α) : (hnp.comp hmn : α → γ) x =
(hnp (hmn x)) | rfl | lemma | ring_hom.comp_apply | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : α →+* β) : f.comp (id α) = f | ext $ λ x, rfl | lemma | ring_hom.comp_id | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →+* β) : (id β).comp f = f | ext $ λ x, rfl | lemma | ring_hom.id_comp | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_def : (1 : α →+* α) = id α | rfl | lemma | ring_hom.one_def | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (f g : α →+* α) : f * g = f.comp g | rfl | lemma | ring_hom.mul_def | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : α →+* α) = _root_.id | rfl | lemma | ring_hom.coe_one | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : α →+* α) : ⇑(f * g) = f ∘ g | rfl | lemma | ring_hom.coe_mul | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : β →+* γ} {f : α →+* β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ring_hom.ext $ hf.forall.2 (ext_iff.1 h), λ h, h ▸ rfl⟩ | lemma | ring_hom.cancel_right | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : β →+* γ} {f₁ f₂ : α →+* β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ring_hom.ext $ λ x, hg $ by rw [← comp_apply, h, comp_apply], λ h, h ▸ rfl⟩ | lemma | ring_hom.cancel_left | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.is_domain [ring α] [is_domain α] [ring β] (f : β →+* α)
(hf : injective f) : is_domain β | begin
haveI := pullback_nonzero f f.map_zero f.map_one,
haveI := is_right_cancel_mul_zero.to_no_zero_divisors α,
haveI := hf.no_zero_divisors f f.map_zero f.map_mul,
exact no_zero_divisors.to_is_domain β,
end | theorem | function.injective.is_domain | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"is_domain",
"is_right_cancel_mul_zero.to_no_zero_divisors",
"no_zero_divisors.to_is_domain",
"pullback_nonzero",
"ring"
] | Pullback `is_domain` instance along an injective function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_ring_hom_of_mul_self_of_two_ne_zero (h : ∀ x, f (x * x) = f x * f x) (h_two : (2 : α) ≠ 0)
(h_one : f 1 = 1) : β →+* α | { map_one' := h_one,
map_mul' := λ x y, begin
have hxy := h (x + y),
rw [mul_add, add_mul, add_mul, f.map_add, f.map_add, f.map_add, f.map_add, h x, h y, add_mul,
mul_add, mul_add, ← sub_eq_zero, add_comm, ← sub_sub, ← sub_sub, ← sub_sub,
mul_comm y x, mul_comm (f y) (f x)] at hxy,
simp only [... | def | add_monoid_hom.mk_ring_hom_of_mul_self_of_two_ne_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [
"mul_comm",
"mul_eq_zero",
"or_iff_not_imp_left",
"two_mul"
] | Make a ring homomorphism from an additive group homomorphism from a commutative ring to an
integral domain that commutes with self multiplication, assumes that two is nonzero and `1` is sent
to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_mk_ring_hom_of_mul_self_of_two_ne_zero (h h_two h_one) :
(f.mk_ring_hom_of_mul_self_of_two_ne_zero h h_two h_one : β → α) = f | rfl | lemma | add_monoid_hom.coe_fn_mk_ring_hom_of_mul_self_of_two_ne_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_monoid_hom_mk_ring_hom_of_mul_self_of_two_ne_zero (h h_two h_one) :
(f.mk_ring_hom_of_mul_self_of_two_ne_zero h h_two h_one : β →+ α) = f | by { ext, refl } | lemma | add_monoid_hom.coe_add_monoid_hom_mk_ring_hom_of_mul_self_of_two_ne_zero | algebra.hom | src/algebra/hom/ring.lean | [
"algebra.group_with_zero.inj_surj",
"algebra.ring.basic",
"algebra.divisibility.basic",
"data.pi.algebra",
"algebra.hom.units",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.is_unit {G} [group G] (g : G) : is_unit g | ⟨⟨g, g⁻¹, mul_inv_self g, inv_mul_self g⟩, rfl⟩ | lemma | group.is_unit | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"group",
"inv_mul_self",
"is_unit",
"mul_inv_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.eq_on_inv {F G N} [division_monoid G] [monoid N] [monoid_hom_class F G N] {x : G}
(hx : is_unit x) (f g : F) (h : f x = g x) : f x⁻¹ = g x⁻¹ | left_inv_eq_right_inv (map_mul_eq_one f hx.inv_mul_cancel) $
h.symm ▸ map_mul_eq_one g $ hx.mul_inv_cancel | lemma | is_unit.eq_on_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"division_monoid",
"is_unit",
"left_inv_eq_right_inv",
"map_mul_eq_one",
"monoid",
"monoid_hom_class"
] | If two homomorphisms from a division monoid to a monoid are equal at a unit `x`, then they are
equal at `x⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_inv {F G M} [group G] [monoid M] [monoid_hom_class F G M] (f g : F) {x : G}
(h : f x = g x) : f x⁻¹ = g x⁻¹ | (group.is_unit x).eq_on_inv f g h | lemma | eq_on_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"group",
"group.is_unit",
"monoid",
"monoid_hom_class"
] | If two homomorphism from a group to a monoid are equal at `x`, then they are equal at `x⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : M →* N) : Mˣ →* Nˣ | monoid_hom.mk'
(λ u, ⟨f u.val, f u.inv,
by rw [← f.map_mul, u.val_inv, f.map_one],
by rw [← f.map_mul, u.inv_val, f.map_one]⟩)
(λ x y, ext (f.map_mul x y)) | def | units.map | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"monoid_hom.mk'"
] | The group homomorphism on units induced by a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (f : M →* N) (x : Mˣ) : ↑(map f x) = f x | rfl | lemma | units.coe_map | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_map_inv (f : M →* N) (u : Mˣ) :
↑(map f u)⁻¹ = f ↑u⁻¹ | rfl | lemma | units.coe_map_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (f : M →* N) (g : N →* P) : map (g.comp f) = (map g).comp (map f) | rfl | lemma | units.map_comp | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (monoid_hom.id M) = monoid_hom.id Mˣ | by ext; refl | lemma | units.map_id | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"map_id",
"monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_hom : Mˣ →* M | ⟨coe, coe_one, coe_mul⟩ | def | units.coe_hom | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [] | Coercion `Mˣ → M` as a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_hom_apply (x : Mˣ) : coe_hom M x = ↑x | rfl | lemma | units.coe_hom_apply | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (u : Mˣ) (n : ℕ) : ((u ^ n : Mˣ) : M) = u ^ n | (units.coe_hom M).map_pow u n | lemma | units.coe_pow | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"map_pow",
"units.coe_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div : ∀ u₁ u₂ : αˣ, ↑(u₁ / u₂) = (u₁ / u₂ : α) | (units.coe_hom α).map_div | lemma | units.coe_div | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"map_div",
"units.coe_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow : ∀ (u : αˣ) (n : ℤ), ((u ^ n : αˣ) : α) = u ^ n | (units.coe_hom α).map_zpow | lemma | units.coe_zpow | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"map_zpow",
"units.coe_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.divp_eq_div (a : α) (u : αˣ) : a /ₚ u = a / u | by rw [div_eq_mul_inv, divp, u.coe_inv] | lemma | divp_eq_div | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"divp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.map_units_inv {F : Type*} [monoid_hom_class F M α] (f : F) (u : units M) :
f ↑u⁻¹ = (f u)⁻¹ | ((f : M →* α).comp (units.coe_hom M)).map_inv u | lemma | map_units_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"map_inv",
"monoid_hom_class",
"units",
"units.coe_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_right (f : M →* N) (g : M → Nˣ) (h : ∀ x, ↑(g x) = f x) :
M →* Nˣ | { to_fun := g,
map_one' := units.ext $ (h 1).symm ▸ f.map_one,
map_mul' := λ x y, units.ext $ by simp only [h, coe_mul, f.map_mul] } | def | units.lift_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"units.ext"
] | If a map `g : M → Nˣ` agrees with a homomorphism `f : M →* N`, then
this map is a monoid homomorphism too. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lift_right {f : M →* N} {g : M → Nˣ}
(h : ∀ x, ↑(g x) = f x) (x) : (lift_right f g h x : N) = f x | h x | lemma | units.coe_lift_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lift_right_inv {f : M →* N} {g : M → Nˣ}
(h : ∀ x, ↑(g x) = f x) (x) : f x * ↑(lift_right f g h x)⁻¹ = 1 | by rw [units.mul_inv_eq_iff_eq_mul, one_mul, coe_lift_right] | lemma | units.mul_lift_right_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"one_mul",
"units.mul_inv_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_right_inv_mul {f : M →* N} {g : M → Nˣ}
(h : ∀ x, ↑(g x) = f x) (x) : ↑(lift_right f g h x)⁻¹ * f x = 1 | by rw [units.inv_mul_eq_iff_eq_mul, mul_one, coe_lift_right] | lemma | units.lift_right_inv_mul | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"mul_one",
"units.inv_mul_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_hom_units {G M : Type*} [group G] [monoid M] (f : G →* M) : G →* Mˣ | units.lift_right f
(λ g, ⟨f g, f g⁻¹, map_mul_eq_one f (mul_inv_self _), map_mul_eq_one f (inv_mul_self _)⟩)
(λ g, rfl) | def | monoid_hom.to_hom_units | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"group",
"inv_mul_self",
"map_mul_eq_one",
"monoid",
"mul_inv_self",
"units.lift_right"
] | If `f` is a homomorphism from a group `G` to a monoid `M`,
then its image lies in the units of `M`,
and `f.to_hom_units` is the corresponding monoid homomorphism from `G` to `Mˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_hom_units {G M : Type*} [group G] [monoid M] (f : G →* M) (g : G) :
(f.to_hom_units g : M) = f g | rfl | lemma | monoid_hom.coe_to_hom_units | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"group",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map [monoid_hom_class F M N] (f : F) {x : M} (h : is_unit x) : is_unit (f x) | by rcases h with ⟨y, rfl⟩; exact (units.map (f : M →* N) y).is_unit | lemma | is_unit.map | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"monoid_hom_class",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_left_inverse [monoid_hom_class F M N] [monoid_hom_class G N M]
{f : F} {x : M} (g : G) (hfg : function.left_inverse g f) (h : is_unit (f x)) :
is_unit x | by simpa only [hfg x] using h.map g | lemma | is_unit.of_left_inverse | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_unit_map_of_left_inverse
[monoid_hom_class F M N] [monoid_hom_class G N M]
{f : F} {x : M} (g : G) (hfg : function.left_inverse g f) :
is_unit (f x) ↔ is_unit x | ⟨of_left_inverse g hfg, map _⟩ | lemma | is_unit_map_of_left_inverse | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_right (f : M →* N) (hf : ∀ x, is_unit (f x)) : M →* Nˣ | units.lift_right f (λ x, (hf x).unit) $ λ x, rfl | def | is_unit.lift_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"units.lift_right"
] | If a homomorphism `f : M →* N` sends each element to an `is_unit`, then it can be lifted
to `f : M →* Nˣ`. See also `units.lift_right` for a computable version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lift_right (f : M →* N) (hf : ∀ x, is_unit (f x)) (x) :
(is_unit.lift_right f hf x : N) = f x | rfl | lemma | is_unit.coe_lift_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"is_unit.lift_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lift_right_inv (f : M →* N) (h : ∀ x, is_unit (f x)) (x) :
f x * ↑(is_unit.lift_right f h x)⁻¹ = 1 | units.mul_lift_right_inv (λ y, rfl) x | lemma | is_unit.mul_lift_right_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"is_unit.lift_right",
"units.mul_lift_right_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_right_inv_mul (f : M →* N) (h : ∀ x, is_unit (f x)) (x) :
↑(is_unit.lift_right f h x)⁻¹ * f x = 1 | units.lift_right_inv_mul (λ y, rfl) x | lemma | is_unit.lift_right_inv_mul | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"is_unit.lift_right",
"units.lift_right_inv_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit' (h : is_unit a) : αˣ | ⟨a, a⁻¹, h.mul_inv_cancel, h.inv_mul_cancel⟩ | def | is_unit.unit' | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit"
] | The element of the group of units, corresponding to an element of a monoid which is a unit. As
opposed to `is_unit.unit`, the inverse is computable and comes from the inversion on `α`. This is
useful to transfer properties of inversion in `units α` to `α`. See also `to_units`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_cancel_left (h : is_unit a) : ∀ b, a * (a⁻¹ * b) = b | h.unit'.mul_inv_cancel_left | lemma | is_unit.mul_inv_cancel_left | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_left (h : is_unit a) : ∀ b, a⁻¹ * (a * b) = b | h.unit'.inv_mul_cancel_left | lemma | is_unit.inv_mul_cancel_left | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"inv_mul_cancel_left",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_right (h : is_unit b) (a : α) :
a * b * b⁻¹ = a | h.unit'.mul_inv_cancel_right _ | lemma | is_unit.mul_inv_cancel_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_right (h : is_unit b) (a : α) :
a * b⁻¹ * b = a | h.unit'.inv_mul_cancel_right _ | lemma | is_unit.inv_mul_cancel_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"inv_mul_cancel_right",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_self (h : is_unit a) : a / a = 1 | by rw [div_eq_mul_inv, h.mul_inv_cancel] | lemma | is_unit.div_self | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"div_self",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_iff_mul_eq (h : is_unit c) : a = b * c⁻¹ ↔ a * c = b | h.unit'.eq_mul_inv_iff_mul_eq | lemma | is_unit.eq_mul_inv_iff_mul_eq | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"eq_mul_inv_iff_mul_eq",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_mul_iff_mul_eq (h : is_unit b) : a = b⁻¹ * c ↔ b * a = c | h.unit'.eq_inv_mul_iff_mul_eq | lemma | is_unit.eq_inv_mul_iff_mul_eq | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"eq_inv_mul_iff_mul_eq",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_iff_eq_mul (h : is_unit a) : a⁻¹ * b = c ↔ b = a * c | h.unit'.inv_mul_eq_iff_eq_mul | lemma | is_unit.inv_mul_eq_iff_eq_mul | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"inv_mul_eq_iff_eq_mul",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_iff_eq_mul (h : is_unit b) : a * b⁻¹ = c ↔ a = c * b | h.unit'.mul_inv_eq_iff_eq_mul | lemma | is_unit.mul_inv_eq_iff_eq_mul | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_inv_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_one (h : is_unit b) : a * b⁻¹ = 1 ↔ a = b | @units.mul_inv_eq_one _ _ h.unit' _ | lemma | is_unit.mul_inv_eq_one | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_inv_eq_one",
"units.mul_inv_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_one (h : is_unit a) : a⁻¹ * b = 1 ↔ a = b | @units.inv_mul_eq_one _ _ h.unit' _ | lemma | is_unit.inv_mul_eq_one | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"inv_mul_eq_one",
"is_unit",
"units.inv_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_eq_inv (h : is_unit b) : a * b = 1 ↔ a = b⁻¹ | @units.mul_eq_one_iff_eq_inv _ _ h.unit' _ | lemma | is_unit.mul_eq_one_iff_eq_inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_eq_one_iff_eq_inv",
"units.mul_eq_one_iff_eq_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_inv_eq (h : is_unit a) : a * b = 1 ↔ a⁻¹ = b | @units.mul_eq_one_iff_inv_eq _ _ h.unit' _ | lemma | is_unit.mul_eq_one_iff_inv_eq | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_eq_one_iff_inv_eq",
"units.mul_eq_one_iff_inv_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel (h : is_unit b) (a : α) : a / b * b = a | by rw [div_eq_mul_inv, h.inv_mul_cancel_right] | lemma | is_unit.div_mul_cancel | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"div_mul_cancel",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel (h : is_unit b) (a : α) : a * b / b = a | by rw [div_eq_mul_inv, h.mul_inv_cancel_right] | lemma | is_unit.mul_div_cancel | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"is_unit",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_div_cancel (h : is_unit a) : a * (1 / a) = 1 | by simp [h] | lemma | is_unit.mul_one_div_cancel | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_one_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_mul_cancel (h : is_unit a) : (1 / a) * a = 1 | by simp [h] | lemma | is_unit.one_div_mul_cancel | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"one_div_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv : is_unit a → is_unit a⁻¹ | by { rintro ⟨u, rfl⟩, rw ←units.coe_inv, exact units.is_unit _ } | lemma | is_unit.inv | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"units.is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div (ha : is_unit a) (hb : is_unit b) : is_unit (a / b) | by { rw div_eq_mul_inv, exact ha.mul hb.inv } | lemma | is_unit.div | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left_inj (h : is_unit c) : a / c = b / c ↔ a = b | by { simp_rw div_eq_mul_inv, exact units.mul_left_inj h.inv.unit' } | lemma | is_unit.div_left_inj | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"div_left_inj",
"is_unit",
"units.mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_iff (h : is_unit b) : a / b = c ↔ a = c * b | by rw [div_eq_mul_inv, h.mul_inv_eq_iff_eq_mul] | lemma | is_unit.div_eq_iff | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_iff",
"div_eq_mul_inv",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_iff (h : is_unit c) : a = b / c ↔ a * c = b | by rw [div_eq_mul_inv, h.eq_mul_inv_iff_mul_eq] | lemma | is_unit.eq_div_iff | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"eq_div_iff",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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