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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|---|---|---|---|---|---|---|---|---|---|---|
of_mul_of {i j} (a : A i) (b : A j) :
of _ i a * of _ j b = of _ (i + j) (graded_monoid.ghas_mul.mul a b) | mul_hom_of_of a b | lemma | direct_sum.of_mul_of | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul (x : ⨁ i, A i) : 1 * x = x | suffices mul_hom A 1 = add_monoid_hom.id (⨁ i, A i),
from add_monoid_hom.congr_fun this x,
begin
apply add_hom_ext, intros i xi,
unfold has_one.one,
rw mul_hom_of_of,
exact of_eq_of_graded_monoid_eq (one_mul $ graded_monoid.mk i xi),
end | lemma | direct_sum.one_mul | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"graded_monoid.mk",
"mul_hom",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one (x : ⨁ i, A i) : x * 1 = x | suffices (mul_hom A).flip 1 = add_monoid_hom.id (⨁ i, A i),
from add_monoid_hom.congr_fun this x,
begin
apply add_hom_ext, intros i xi,
unfold has_one.one,
rw [flip_apply, mul_hom_of_of],
exact of_eq_of_graded_monoid_eq (mul_one $ graded_monoid.mk i xi),
end | lemma | direct_sum.mul_one | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"graded_monoid.mk",
"mul_hom",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_assoc (a b c : ⨁ i, A i) : a * b * c = a * (b * c) | suffices (mul_hom A).comp_hom.comp (mul_hom A) -- `λ a b c, a * b * c` as a bundled hom
= (add_monoid_hom.comp_hom flip_hom $ -- `λ a b c, a * (b * c)` as a bundled hom
(mul_hom A).flip.comp_hom.comp (mul_hom A)).flip,
from add_monoid_hom.congr_fun (add_monoid_hom.congr_fun... | lemma | direct_sum.mul_assoc | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"graded_monoid.mk",
"mul_assoc",
"mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring : semiring (⨁ i, A i) | { one := 1,
mul := (*),
zero := 0,
add := (+),
one_mul := one_mul A,
mul_one := mul_one A,
mul_assoc := mul_assoc A,
nat_cast := λ n, of _ _ (gsemiring.nat_cast n),
nat_cast_zero := by rw [gsemiring.nat_cast_zero, map_zero],
nat_cast_succ := λ n, by { rw [gsemiring.nat_cast_succ, map_add], refl },
.... | instance | direct_sum.semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"mul_assoc",
"mul_one",
"one_mul",
"semiring"
] | The `semiring` structure derived from `gsemiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_pow {i} (a : A i) (n : ℕ) :
of _ i a ^ n = of _ (n • i) (graded_monoid.gmonoid.gnpow _ a) | begin
induction n with n,
{ exact of_eq_of_graded_monoid_eq (pow_zero $ graded_monoid.mk _ a).symm, },
{ rw [pow_succ, n_ih, of_mul_of],
exact of_eq_of_graded_monoid_eq (pow_succ (graded_monoid.mk _ a) n).symm, },
end | lemma | direct_sum.of_pow | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"graded_monoid.mk",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_list_dprod {α} (l : list α) (fι : α → ι) (fA : Π a, A (fι a)) :
of A _ (l.dprod fι fA) = (l.map $ λ a, of A (fι a) (fA a)).prod | begin
induction l,
{ simp only [list.map_nil, list.prod_nil, list.dprod_nil],
refl },
{ simp only [list.map_cons, list.prod_cons, list.dprod_cons, ←l_ih, direct_sum.of_mul_of],
refl },
end | lemma | direct_sum.of_list_dprod | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"direct_sum.of_mul_of",
"list.dprod_cons",
"list.dprod_nil",
"list.map_nil",
"list.prod_cons",
"list.prod_nil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
list_prod_of_fn_of_eq_dprod (n : ℕ) (fι : fin n → ι) (fA : Π a, A (fι a)) :
(list.of_fn $ λ a, of A (fι a) (fA a)).prod = of A _ ((list.fin_range n).dprod fι fA) | by rw [list.of_fn_eq_map, of_list_dprod] | lemma | direct_sum.list_prod_of_fn_of_eq_dprod | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"list.fin_range",
"list.of_fn",
"list.of_fn_eq_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_dfinsupp_sum [Π (i : ι) (x : A i), decidable (x ≠ 0)] (a a' : ⨁ i, A i) :
a * a' = a.sum (λ i ai, a'.sum $ λ j aj, direct_sum.of _ _ $ graded_monoid.ghas_mul.mul ai aj) | begin
change mul_hom _ a a' = _,
simpa only [mul_hom, to_add_monoid, dfinsupp.lift_add_hom_apply, dfinsupp.sum_add_hom_apply,
add_monoid_hom.dfinsupp_sum_apply, flip_apply, add_monoid_hom.dfinsupp_sum_add_hom_apply],
end | lemma | direct_sum.mul_eq_dfinsupp_sum | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_monoid_hom.dfinsupp_sum_add_hom_apply",
"dfinsupp.sum_add_hom_apply",
"direct_sum.of",
"mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_sum_support_ghas_mul
[Π (i : ι) (x : A i), decidable (x ≠ 0)] (a a' : ⨁ i, A i) :
a * a' =
∑ ij in dfinsupp.support a ×ˢ dfinsupp.support a',
direct_sum.of _ _ (graded_monoid.ghas_mul.mul (a ij.fst) (a' ij.snd)) | by simp only [mul_eq_dfinsupp_sum, dfinsupp.sum, finset.sum_product] | lemma | direct_sum.mul_eq_sum_support_ghas_mul | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"dfinsupp.support",
"direct_sum.of"
] | A heavily unfolded version of the definition of multiplication | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_comm (a b : ⨁ i, A i) : a * b = b * a | suffices mul_hom A = (mul_hom A).flip,
from add_monoid_hom.congr_fun (add_monoid_hom.congr_fun this a) b,
begin
apply add_hom_ext, intros ai ax, apply add_hom_ext, intros bi bx,
rw [add_monoid_hom.flip_apply, mul_hom_of_of, mul_hom_of_of],
exact of_eq_of_graded_monoid_eq (gcomm_semiring.mul_comm ⟨ai, ax⟩ ⟨bi, b... | lemma | direct_sum.mul_comm | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"mul_comm",
"mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring : comm_semiring (⨁ i, A i) | { one := 1,
mul := (*),
zero := 0,
add := (+),
mul_comm := mul_comm A,
..direct_sum.semiring _, } | instance | direct_sum.comm_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"comm_semiring",
"direct_sum.semiring",
"mul_comm"
] | The `comm_semiring` structure derived from `gcomm_semiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_assoc_ring : non_unital_non_assoc_ring (⨁ i, A i) | { mul := (*),
zero := 0,
add := (+),
neg := has_neg.neg,
..(direct_sum.non_unital_non_assoc_semiring _),
..(direct_sum.add_comm_group _), } | instance | direct_sum.non_assoc_ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"non_assoc_ring",
"non_unital_non_assoc_ring"
] | The `ring` derived from `gsemiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring : ring (⨁ i, A i) | { one := 1,
mul := (*),
zero := 0,
add := (+),
neg := has_neg.neg,
int_cast := λ z, of _ _ (gring.int_cast z),
int_cast_of_nat := λ z, congr_arg _ $ gring.int_cast_of_nat _,
int_cast_neg_succ_of_nat := λ z,
(congr_arg _ $ gring.int_cast_neg_succ_of_nat _).trans (map_neg _ _),
..(direct_sum.semiring ... | instance | direct_sum.ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"direct_sum.semiring",
"ring"
] | The `ring` derived from `gsemiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring : comm_ring (⨁ i, A i) | { one := 1,
mul := (*),
zero := 0,
add := (+),
neg := has_neg.neg,
..(direct_sum.ring _),
..(direct_sum.comm_semiring _), } | instance | direct_sum.comm_ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"comm_ring",
"direct_sum.comm_semiring",
"direct_sum.ring"
] | The `comm_ring` derived from `gcomm_semiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_zero_one : of _ 0 (1 : A 0) = 1 | rfl | lemma | direct_sum.of_zero_one | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_zero_smul {i} (a : A 0) (b : A i) : of _ _ (a • b) = of _ _ a * of _ _ b | (of_eq_of_graded_monoid_eq (graded_monoid.mk_zero_smul a b)).trans (of_mul_of _ _).symm | lemma | direct_sum.of_zero_smul | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"graded_monoid.mk_zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_zero_mul (a b : A 0) : of _ 0 (a * b) = of _ 0 a * of _ 0 b | of_zero_smul A a b | lemma | direct_sum.of_zero_mul | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
grade_zero.non_unital_non_assoc_semiring : non_unital_non_assoc_semiring (A 0) | function.injective.non_unital_non_assoc_semiring (of A 0) dfinsupp.single_injective
(of A 0).map_zero (of A 0).map_add (of_zero_mul A) (λ x n, dfinsupp.single_smul n x) | instance | direct_sum.grade_zero.non_unital_non_assoc_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"dfinsupp.single_injective",
"dfinsupp.single_smul",
"function.injective.non_unital_non_assoc_semiring",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
grade_zero.smul_with_zero (i : ι) : smul_with_zero (A 0) (A i) | begin
letI := smul_with_zero.comp_hom (⨁ i, A i) (of A 0).to_zero_hom,
refine dfinsupp.single_injective.smul_with_zero (of A i).to_zero_hom (of_zero_smul A),
end | instance | direct_sum.grade_zero.smul_with_zero | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"smul_with_zero",
"smul_with_zero.comp_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_zero_pow (a : A 0) : ∀ n : ℕ, of _ 0 (a ^ n) = of _ 0 a ^ n | | 0 := by rw [pow_zero, pow_zero, direct_sum.of_zero_one]
| (n + 1) := by rw [pow_succ, pow_succ, of_zero_mul, of_zero_pow] | lemma | direct_sum.of_zero_pow | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"direct_sum.of_zero_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_nat_cast (n : ℕ) : of A 0 n = n | rfl | lemma | direct_sum.of_nat_cast | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
grade_zero.semiring : semiring (A 0) | function.injective.semiring (of A 0) dfinsupp.single_injective
(of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)
(of A 0).map_nsmul (λ x n, of_zero_pow _ _ _) (of_nat_cast A) | instance | direct_sum.grade_zero.semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"dfinsupp.single_injective",
"function.injective.semiring",
"map_nsmul",
"semiring"
] | The `semiring` structure derived from `gsemiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_zero_ring_hom : A 0 →+* (⨁ i, A i) | { map_one' := of_zero_one A, map_mul' := of_zero_mul A, ..(of _ 0) } | def | direct_sum.of_zero_ring_hom | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | `of A 0` is a `ring_hom`, using the `direct_sum.grade_zero.semiring` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.module {i} : module (A 0) (A i) | begin
letI := module.comp_hom (⨁ i, A i) (of_zero_ring_hom A),
exact dfinsupp.single_injective.module (A 0) (of A i) (λ a, of_zero_smul A a),
end | instance | direct_sum.grade_zero.module | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"module",
"module.comp_hom"
] | Each grade `A i` derives a `A 0`-module structure from `gsemiring A`. Note that this results
in an overall `module (A 0) (⨁ i, A i)` structure via `direct_sum.module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.comm_semiring : comm_semiring (A 0) | function.injective.comm_semiring (of A 0) dfinsupp.single_injective
(of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)
(λ x n, dfinsupp.single_smul n x) (λ x n, of_zero_pow _ _ _) (of_nat_cast A) | instance | direct_sum.grade_zero.comm_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"comm_semiring",
"dfinsupp.single_injective",
"dfinsupp.single_smul",
"function.injective.comm_semiring"
] | The `comm_semiring` structure derived from `gcomm_semiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.non_unital_non_assoc_ring : non_unital_non_assoc_ring (A 0) | function.injective.non_unital_non_assoc_ring (of A 0) dfinsupp.single_injective
(of A 0).map_zero (of A 0).map_add (of_zero_mul A)
(of A 0).map_neg (of A 0).map_sub
(λ x n, begin
letI : Π i, distrib_mul_action ℕ (A i) := λ i, infer_instance,
exact dfinsupp.single_smul n x
end)
(λ x n, begin
letI :... | instance | direct_sum.grade_zero.non_unital_non_assoc_ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"dfinsupp.single_injective",
"dfinsupp.single_smul",
"distrib_mul_action",
"function.injective.non_unital_non_assoc_ring",
"non_unital_non_assoc_ring"
] | The `non_unital_non_assoc_ring` derived from `gnon_unital_non_assoc_semiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_int_cast (n : ℤ) : of A 0 n = n | rfl | lemma | direct_sum.of_int_cast | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
grade_zero.ring : ring (A 0) | function.injective.ring (of A 0) dfinsupp.single_injective
(of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)
(of A 0).map_neg (of A 0).map_sub
(λ x n, begin
letI : Π i, distrib_mul_action ℕ (A i) := λ i, infer_instance,
exact dfinsupp.single_smul n x
end)
(λ x n, begin
letI : Π i,... | instance | direct_sum.grade_zero.ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"dfinsupp.single_injective",
"dfinsupp.single_smul",
"distrib_mul_action",
"function.injective.ring",
"ring"
] | The `ring` derived from `gsemiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
grade_zero.comm_ring : comm_ring (A 0) | function.injective.comm_ring (of A 0) dfinsupp.single_injective
(of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)
(of A 0).map_neg (of A 0).map_sub
(λ x n, begin
letI : Π i, distrib_mul_action ℕ (A i) := λ i, infer_instance,
exact dfinsupp.single_smul n x
end)
(λ x n, begin
letI :... | instance | direct_sum.grade_zero.comm_ring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"comm_ring",
"dfinsupp.single_injective",
"dfinsupp.single_smul",
"distrib_mul_action",
"function.injective.comm_ring"
] | The `comm_ring` derived from `gcomm_semiring A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_ext' ⦃F G : (⨁ i, A i) →+* R⦄
(h : ∀ i, (↑F : _ →+ R).comp (of A i) = (↑G : _ →+ R).comp (of A i)) : F = G | ring_hom.coe_add_monoid_hom_injective $ direct_sum.add_hom_ext' h | lemma | direct_sum.ring_hom_ext' | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"direct_sum.add_hom_ext'",
"ring_hom.coe_add_monoid_hom_injective"
] | If two ring homomorphisms from `⨁ i, A i` are equal on each `of A i y`,
then they are equal.
See note [partially-applied ext lemmas]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_ext ⦃f g : (⨁ i, A i) →+* R⦄ (h : ∀ i x, f (of A i x) = g (of A i x)) :
f = g | ring_hom_ext' $ λ i, add_monoid_hom.ext $ h i | lemma | direct_sum.ring_hom_ext | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"ring_hom_ext"
] | Two `ring_hom`s out of a direct sum are equal if they agree on the generators. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_semiring
(f : Π i, A i →+ R) (hone : f _ (graded_monoid.ghas_one.one) = 1)
(hmul : ∀ {i j} (ai : A i) (aj : A j), f _ (graded_monoid.ghas_mul.mul ai aj) = f _ ai * f _ aj) :
(⨁ i, A i) →+* R | { to_fun := to_add_monoid f,
map_one' := begin
change (to_add_monoid f) (of _ 0 _) = 1,
rw to_add_monoid_of,
exact hone
end,
map_mul' := begin
rw (to_add_monoid f).map_mul_iff,
ext xi xv yi yv : 4,
show to_add_monoid f (of A xi xv * of A yi yv) =
to_add_monoid f (of A xi xv) * to_... | def | direct_sum.to_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | A family of `add_monoid_hom`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`
describes a `ring_hom`s on `⨁ i, A i`. This is a stronger version of `direct_sum.to_monoid`.
Of particular interest is the case when `A i` are bundled subojects, `f` is the family of
coercions such as `add_submonoid.subtyp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_semiring_of (f : Π i, A i →+ R) (hone hmul) (i : ι) (x : A i) :
to_semiring f hone hmul (of _ i x) = f _ x | to_add_monoid_of f i x | lemma | direct_sum.to_semiring_of | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_semiring_coe_add_monoid_hom (f : Π i, A i →+ R) (hone hmul):
(to_semiring f hone hmul : (⨁ i, A i) →+ R) = to_add_monoid f | rfl | lemma | direct_sum.to_semiring_coe_add_monoid_hom | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_ring_hom :
{f : Π {i}, A i →+ R //
f (graded_monoid.ghas_one.one) = 1 ∧
∀ {i j} (ai : A i) (aj : A j), f (graded_monoid.ghas_mul.mul ai aj) = f ai * f aj} ≃
((⨁ i, A i) →+* R) | { to_fun := λ f, to_semiring (λ _, f.1) f.2.1 (λ _ _, f.2.2),
inv_fun := λ F,
⟨λ i, (F : (⨁ i, A i) →+ R).comp (of _ i), begin
simp only [add_monoid_hom.comp_apply, ring_hom.coe_add_monoid_hom],
rw ←F.map_one,
refl
end, λ i j ai aj, begin
simp only [add_monoid_hom.comp_apply, ring_hom.... | def | direct_sum.lift_ring_hom | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"direct_sum.to_add_monoid_of",
"inv_fun",
"ring_hom.coe_add_monoid_hom",
"ring_hom.coe_add_monoid_hom_injective",
"ring_hom.coe_add_monoid_hom_mk"
] | Families of `add_monoid_hom`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`
are isomorphic to `ring_hom`s on `⨁ i, A i`. This is a stronger version of `dfinsupp.lift_add_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring
{R : Type*} [add_monoid ι] [non_unital_non_assoc_semiring R] :
direct_sum.gnon_unital_non_assoc_semiring (λ i : ι, R) | { mul_zero := λ i j, mul_zero,
zero_mul := λ i j, zero_mul,
mul_add := λ i j, mul_add,
add_mul := λ i j, add_mul,
..has_mul.ghas_mul ι } | instance | non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_monoid",
"direct_sum.gnon_unital_non_assoc_semiring",
"has_mul.ghas_mul",
"mul_zero",
"non_unital_non_assoc_semiring",
"zero_mul"
] | A direct sum of copies of a `semiring` inherits the multiplication structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semiring.direct_sum_gsemiring {R : Type*} [add_monoid ι] [semiring R] :
direct_sum.gsemiring (λ i : ι, R) | { nat_cast := λ n, n,
nat_cast_zero := nat.cast_zero,
nat_cast_succ := nat.cast_succ,
..non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring ι,
..monoid.gmonoid ι } | instance | semiring.direct_sum_gsemiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_monoid",
"direct_sum.gsemiring",
"monoid.gmonoid",
"nat.cast_succ",
"nat.cast_zero",
"non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring",
"semiring"
] | A direct sum of copies of a `semiring` inherits the multiplication structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semiring.direct_sum_gcomm_semiring {R : Type*} [add_comm_monoid ι] [comm_semiring R] :
direct_sum.gcomm_semiring (λ i : ι, R) | { ..comm_monoid.gcomm_monoid ι, ..semiring.direct_sum_gsemiring ι } | instance | comm_semiring.direct_sum_gcomm_semiring | algebra.direct_sum | src/algebra/direct_sum/ring.lean | [
"algebra.graded_monoid",
"algebra.direct_sum.basic"
] | [
"add_comm_monoid",
"comm_monoid.gcomm_monoid",
"comm_semiring",
"direct_sum.gcomm_semiring",
"semiring.direct_sum_gsemiring"
] | A direct sum of copies of a `comm_semiring` inherits the commutative multiplication structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semigroup_has_dvd : has_dvd α | has_dvd.mk (λ a b, ∃ c, b = a * c) | instance | semigroup_has_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [] | There are two possible conventions for divisibility, which coincide in a `comm_monoid`.
This matches the convention for ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd.intro (c : α) (h : a * c = b) : a ∣ b | exists.intro c h^.symm | theorem | dvd.intro | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_mul_right_of_dvd (h : a ∣ b) : ∃ c, b = a * c | h | theorem | exists_eq_mul_right_of_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd.elim {P : Prop} {a b : α} (H₁ : a ∣ b) (H₂ : ∀ c, b = a * c → P) : P | exists.elim H₁ H₂ | theorem | dvd.elim | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_trans : a ∣ b → b ∣ c → a ∣ c | | ⟨d, h₁⟩ ⟨e, h₂⟩ := ⟨d * e, h₁ ▸ h₂.trans $ mul_assoc a d e⟩ | theorem | dvd_trans | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_mul_right (a b : α) : a ∣ a * b | dvd.intro b rfl | theorem | dvd_mul_right | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.intro"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_mul_of_dvd_left (h : a ∣ b) (c : α) : a ∣ b * c | h.trans (dvd_mul_right b c) | theorem | dvd_mul_of_dvd_left | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_mul_right_dvd (h : a * b ∣ c) : a ∣ c | (dvd_mul_right a b).trans h | theorem | dvd_of_mul_right_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_dvd {F : Type*} [mul_hom_class F M N] (f : F) {a b} : a ∣ b → f a ∣ f b | | ⟨c, h⟩ := ⟨f c, h.symm ▸ map_mul f a c⟩ | lemma | map_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"map_mul",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.map_dvd (f : M →ₙ* N) {a b} : a ∣ b → f a ∣ f b | map_dvd f | lemma | mul_hom.map_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"map_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_dvd (f : M →* N) {a b} : a ∣ b → f a ∣ f b | map_dvd f | lemma | monoid_hom.map_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"map_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_refl (a : α) : a ∣ a | dvd.intro 1 (mul_one a) | theorem | dvd_refl | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.intro",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_rfl : ∀ {a : α}, a ∣ a | dvd_refl | theorem | dvd_rfl | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_dvd (a : α) : 1 ∣ a | dvd.intro a (one_mul a) | theorem | one_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.intro",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_eq (h : a = b) : a ∣ b | by rw h | lemma | dvd_of_eq | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd.intro_left (c : α) (h : c * a = b) : a ∣ b | dvd.intro _ (begin rewrite mul_comm at h, apply h end) | theorem | dvd.intro_left | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.intro",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_mul_left_of_dvd (h : a ∣ b) : ∃ c, b = c * a | dvd.elim h (assume c, assume H1 : b = a * c, exists.intro c (eq.trans H1 (mul_comm a c))) | theorem | exists_eq_mul_left_of_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.elim",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_iff_exists_eq_mul_left : a ∣ b ↔ ∃ c, b = c * a | ⟨exists_eq_mul_left_of_dvd, by { rintro ⟨c, rfl⟩, exact ⟨c, mul_comm _ _⟩, }⟩ | lemma | dvd_iff_exists_eq_mul_left | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd.elim_left {P : Prop} (h₁ : a ∣ b) (h₂ : ∀ c, b = c * a → P) : P | exists.elim (exists_eq_mul_left_of_dvd h₁) (assume c, assume h₃ : b = c * a, h₂ c h₃) | theorem | dvd.elim_left | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"exists_eq_mul_left_of_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_mul_left (a b : α) : a ∣ b * a | dvd.intro b (mul_comm a b) | theorem | dvd_mul_left | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.intro",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_mul_of_dvd_right (h : a ∣ b) (c : α) : a ∣ c * b | begin rw mul_comm, exact h.mul_right _ end | theorem | dvd_mul_of_dvd_right | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_dvd_mul : ∀ {a b c d : α}, a ∣ b → c ∣ d → a * c ∣ b * d | | a ._ c ._ ⟨e, rfl⟩ ⟨f, rfl⟩ := ⟨e * f, by simp⟩ | theorem | mul_dvd_mul | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_mul_left_dvd (h : a * b ∣ c) : b ∣ c | dvd.elim h (λ d ceq, dvd.intro (a * d) (by simp [ceq])) | theorem | dvd_of_mul_left_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd.elim",
"dvd.intro"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_dvd_mul_left (a : α) {b c : α} (h : b ∣ c) : a * b ∣ a * c | mul_dvd_mul (dvd_refl a) h | theorem | mul_dvd_mul_left | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd_refl",
"mul_dvd_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_dvd_mul_right (h : a ∣ b) (c : α) : a * c ∣ b * c | mul_dvd_mul h (dvd_refl c) | theorem | mul_dvd_mul_right | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"dvd_refl",
"mul_dvd_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_dvd_pow_of_dvd {a b : α} (h : a ∣ b) : ∀ n : ℕ, a ^ n ∣ b ^ n | | 0 := by rw [pow_zero, pow_zero]
| (n+1) := by { rw [pow_succ, pow_succ], exact mul_dvd_mul h (pow_dvd_pow_of_dvd n) } | theorem | pow_dvd_pow_of_dvd | algebra.divisibility | src/algebra/divisibility/basic.lean | [
"algebra.hom.group"
] | [
"mul_dvd_mul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_dvd : ↑u ∣ a | ⟨↑u⁻¹ * a, by simp⟩ | lemma | units.coe_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [] | Elements of the unit group of a monoid represented as elements of the monoid
divide any element of the monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_mul_right : a ∣ b * u ↔ a ∣ b | iff.intro
(assume ⟨c, eq⟩, ⟨c * ↑u⁻¹, by rw [← mul_assoc, ← eq, units.mul_inv_cancel_right]⟩)
(assume ⟨c, eq⟩, eq.symm ▸ (dvd_mul_right _ _).mul_right _) | lemma | units.dvd_mul_right | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"dvd_mul_right",
"mul_assoc",
"units.mul_inv_cancel_right"
] | In a monoid, an element `a` divides an element `b` iff `a` divides all
associates of `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right_dvd : a * u ∣ b ↔ a ∣ b | iff.intro
(λ ⟨c, eq⟩, ⟨↑u * c, eq.trans (mul_assoc _ _ _)⟩)
(λ h, dvd_trans (dvd.intro ↑u⁻¹ (by rw [mul_assoc, u.mul_inv, mul_one])) h) | lemma | units.mul_right_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"dvd.intro",
"dvd_trans",
"mul_assoc",
"mul_one"
] | In a monoid, an element `a` divides an element `b` iff all associates of `a` divide `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_mul_left : a ∣ u * b ↔ a ∣ b | by { rw mul_comm, apply dvd_mul_right } | lemma | units.dvd_mul_left | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"dvd_mul_left",
"dvd_mul_right",
"mul_comm"
] | In a commutative monoid, an element `a` divides an element `b` iff `a` divides all left
associates of `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_dvd : ↑u * a ∣ b ↔ a ∣ b | by { rw mul_comm, apply mul_right_dvd } | lemma | units.mul_left_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"mul_comm"
] | In a commutative monoid, an element `a` divides an element `b` iff all
left associates of `a` divide `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd : u ∣ a | by { rcases hu with ⟨u, rfl⟩, apply units.coe_dvd, } | lemma | is_unit.dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"units.coe_dvd"
] | Units of a monoid divide any element of the monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_mul_right : a ∣ b * u ↔ a ∣ b | by { rcases hu with ⟨u, rfl⟩, apply units.dvd_mul_right, } | lemma | is_unit.dvd_mul_right | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"dvd_mul_right",
"units.dvd_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_dvd : a * u ∣ b ↔ a ∣ b | by { rcases hu with ⟨u, rfl⟩, apply units.mul_right_dvd, } | lemma | is_unit.mul_right_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"units.mul_right_dvd"
] | In a monoid, an element a divides an element b iff all associates of `a` divide `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dvd_mul_left : a ∣ u * b ↔ a ∣ b | by { rcases hu with ⟨u, rfl⟩, apply units.dvd_mul_left, } | lemma | is_unit.dvd_mul_left | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"dvd_mul_left",
"units.dvd_mul_left"
] | In a commutative monoid, an element `a` divides an element `b` iff `a` divides all left
associates of `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_dvd : u * a ∣ b ↔ a ∣ b | by { rcases hu with ⟨u, rfl⟩, apply units.mul_left_dvd, } | lemma | is_unit.mul_left_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"units.mul_left_dvd"
] | In a commutative monoid, an element `a` divides an element `b` iff all
left associates of `a` divide `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff_dvd_one {x : α} : is_unit x ↔ x ∣ 1 | ⟨is_unit.dvd, λ ⟨y, h⟩, ⟨⟨x, y, h.symm, by rw [h, mul_comm]⟩, rfl⟩⟩ | theorem | is_unit_iff_dvd_one | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"is_unit",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_forall_dvd {x : α} :
is_unit x ↔ ∀ y, x ∣ y | is_unit_iff_dvd_one.trans ⟨λ h y, h.trans (one_dvd _), λ h, h _⟩ | theorem | is_unit_iff_forall_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"is_unit",
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_dvd_unit {x y : α}
(xy : x ∣ y) (hu : is_unit y) : is_unit x | is_unit_iff_dvd_one.2 $ xy.trans $ is_unit_iff_dvd_one.1 hu | theorem | is_unit_of_dvd_unit | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_dvd_one : ∀a ∣ 1, is_unit (a:α) | | a ⟨b, eq⟩ := ⟨units.mk_of_mul_eq_one a b eq.symm, rfl⟩ | lemma | is_unit_of_dvd_one | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_unit_of_not_is_unit_dvd {a b : α} (ha : ¬is_unit a) (hb : a ∣ b) :
¬ is_unit b | mt (is_unit_of_dvd_unit hb) ha | lemma | not_is_unit_of_not_is_unit_dvd | algebra.divisibility | src/algebra/divisibility/units.lean | [
"algebra.divisibility.basic",
"algebra.group.units"
] | [
"is_unit",
"is_unit_of_dvd_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel_left {a : R} (b) (a0 : a ≠ 0) : a * b / a = b | eq.symm $ eq_of_sub_eq_zero $ classical.by_contradiction $ λ h,
begin
have := mul_left_not_lt a h,
rw [mul_sub, sub_eq_iff_eq_add'.2 (div_add_mod (a*b) a).symm] at this,
exact this (mod_lt _ a0)
end | lemma | euclidean_domain.mul_div_cancel_left | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel (a) {b : R} (b0 : b ≠ 0) : a * b / b = a | by { rw mul_comm, exact mul_div_cancel_left a b0 } | lemma | euclidean_domain.mul_div_cancel | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"mul_comm",
"mul_div_cancel",
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_eq_zero {a b : R} : a % b = 0 ↔ b ∣ a | ⟨λ h, by { rw [← div_add_mod a b, h, add_zero], exact dvd_mul_right _ _ },
λ ⟨c, e⟩, begin
rw [e, ← add_left_cancel_iff, div_add_mod, add_zero],
haveI := classical.dec,
by_cases b0 : b = 0,
{ simp only [b0, zero_mul] },
{ rw [mul_div_cancel_left _ b0] }
end⟩ | lemma | euclidean_domain.mod_eq_zero | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"classical.dec",
"dvd_mul_right",
"mul_div_cancel_left",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_self (a : R) : a % a = 0 | mod_eq_zero.2 dvd_rfl | lemma | euclidean_domain.mod_self | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_mod_iff {a b c : R} (h : c ∣ b) : c ∣ a % b ↔ c ∣ a | by rw [←dvd_add_right (h.mul_right _), div_add_mod] | lemma | euclidean_domain.dvd_mod_iff | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_one (a : R) : a % 1 = 0 | mod_eq_zero.2 (one_dvd _) | lemma | euclidean_domain.mod_one | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"one_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mod (b : R) : 0 % b = 0 | mod_eq_zero.2 (dvd_zero _) | lemma | euclidean_domain.zero_mod | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"dvd_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_div {a : R} : 0 / a = 0 | classical.by_cases
(λ a0 : a = 0, a0.symm ▸ div_zero 0)
(λ a0, by simpa only [zero_mul] using mul_div_cancel 0 a0) | lemma | euclidean_domain.zero_div | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"div_zero",
"mul_div_cancel",
"zero_div",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_self {a : R} (a0 : a ≠ 0) : a / a = 1 | by simpa only [one_mul] using mul_div_cancel 1 a0 | lemma | euclidean_domain.div_self | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"div_self",
"mul_div_cancel",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_of_mul_eq_left {a b c : R} (hb : b ≠ 0) (h : a * b = c) : a = c / b | by rw [← h, mul_div_cancel _ hb] | lemma | euclidean_domain.eq_div_of_mul_eq_left | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_of_mul_eq_right {a b c : R} (ha : a ≠ 0) (h : a * b = c) : b = c / a | by rw [← h, mul_div_cancel_left _ ha] | lemma | euclidean_domain.eq_div_of_mul_eq_right | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_assoc (x : R) {y z : R} (h : z ∣ y) : x * y / z = x * (y / z) | begin
classical, by_cases hz : z = 0,
{ subst hz, rw [div_zero, div_zero, mul_zero] },
rcases h with ⟨p, rfl⟩,
rw [mul_div_cancel_left _ hz, mul_left_comm, mul_div_cancel_left _ hz]
end | theorem | euclidean_domain.mul_div_assoc | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"div_zero",
"mul_div_assoc",
"mul_div_cancel_left",
"mul_left_comm",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel' {a b : R} (hb : b ≠ 0) (hab : b ∣ a) : b * (a / b) = a | by rw [←mul_div_assoc _ hab, mul_div_cancel_left _ hb] | lemma | euclidean_domain.mul_div_cancel' | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"mul_div_cancel'",
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_one (p : R) : p / 1 = p | (euclidean_domain.eq_div_of_mul_eq_left (one_ne_zero' R) (mul_one p)).symm | lemma | euclidean_domain.div_one | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"div_one",
"euclidean_domain.eq_div_of_mul_eq_left",
"mul_one",
"one_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_dvd_of_dvd {p q : R} (hpq : q ∣ p) :
p / q ∣ p | begin
by_cases hq : q = 0,
{ rw [hq, zero_dvd_iff] at hpq,
rw hpq,
exact dvd_zero _ },
use q,
rw [mul_comm, ← euclidean_domain.mul_div_assoc _ hpq, mul_comm,
euclidean_domain.mul_div_cancel _ hq]
end | lemma | euclidean_domain.div_dvd_of_dvd | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"dvd_zero",
"euclidean_domain.mul_div_assoc",
"euclidean_domain.mul_div_cancel",
"mul_comm",
"zero_dvd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_div_of_mul_dvd {a b c : R} (h : a * b ∣ c) : b ∣ c / a | begin
rcases eq_or_ne a 0 with rfl | ha,
{ simp only [div_zero, dvd_zero] },
rcases h with ⟨d, rfl⟩,
refine ⟨d, _⟩,
rw [mul_assoc, mul_div_cancel_left _ ha]
end | lemma | euclidean_domain.dvd_div_of_mul_dvd | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"div_zero",
"dvd_zero",
"eq_or_ne",
"mul_assoc",
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_zero_right (a : R) : gcd a 0 = a | by { rw gcd, split_ifs; simp only [h, zero_mod, gcd_zero_left] } | theorem | euclidean_domain.gcd_zero_right | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"gcd_zero_left",
"gcd_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_val (a b : R) : gcd a b = gcd (b % a) a | by { rw gcd, split_ifs; [simp only [h, mod_zero, gcd_zero_right], refl]} | theorem | euclidean_domain.gcd_val | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"gcd_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd (a b : R) : gcd a b ∣ a ∧ gcd a b ∣ b | gcd.induction a b
(λ b, by { rw gcd_zero_left, exact ⟨dvd_zero _, dvd_rfl⟩ })
(λ a b aneq ⟨IH₁, IH₂⟩, by { rw gcd_val,
exact ⟨IH₂, (dvd_mod_iff IH₂).1 IH₁⟩ }) | theorem | euclidean_domain.gcd_dvd | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [
"gcd_zero_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_left (a b : R) : gcd a b ∣ a | (gcd_dvd a b).left | theorem | euclidean_domain.gcd_dvd_left | algebra.euclidean_domain | src/algebra/euclidean_domain/basic.lean | [
"algebra.euclidean_domain.defs",
"algebra.ring.divisibility",
"algebra.ring.regular",
"algebra.group_with_zero.divisibility",
"algebra.ring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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