statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
norm_sq_inv : norm_sq a⁻¹ = (norm_sq a)⁻¹ | map_inv₀ norm_sq _ | lemma | quaternion.norm_sq_inv | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"map_inv₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_div : norm_sq (a / b) = norm_sq a / norm_sq b | map_div₀ norm_sq a b | lemma | quaternion.norm_sq_div | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"map_div₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_zpow (z : ℤ) : norm_sq (a ^ z) = norm_sq a ^ z | map_zpow₀ norm_sq a z | lemma | quaternion.norm_sq_zpow | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"map_zpow₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_rat_cast (q : ℚ) : norm_sq (q : ℍ[R]) = q^2 | by rw [←coe_rat_cast, norm_sq_coe] | lemma | quaternion.norm_sq_rat_cast | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_four [infinite R] : #R ^ 4 = #R | power_nat_eq (aleph_0_le_mk R) $ by simp | theorem | cardinal.pow_four | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_quaternion_algebra : #ℍ[R, c₁, c₂] = #R ^ 4 | by { rw mk_congr (quaternion_algebra.equiv_prod c₁ c₂), simp only [mk_prod, lift_id], ring } | lemma | cardinal.mk_quaternion_algebra | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"quaternion_algebra.equiv_prod",
"ring"
] | The cardinality of a quaternion algebra, as a type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_quaternion_algebra_of_infinite [infinite R] : #ℍ[R, c₁, c₂] = #R | by rw [mk_quaternion_algebra, pow_four] | lemma | cardinal.mk_quaternion_algebra_of_infinite | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_univ_quaternion_algebra : #(set.univ : set ℍ[R, c₁, c₂]) = #R ^ 4 | by rw [mk_univ, mk_quaternion_algebra] | lemma | cardinal.mk_univ_quaternion_algebra | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | The cardinality of a quaternion algebra, as a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_univ_quaternion_algebra_of_infinite [infinite R] :
#(set.univ : set ℍ[R, c₁, c₂]) = #R | by rw [mk_univ_quaternion_algebra, pow_four] | lemma | cardinal.mk_univ_quaternion_algebra_of_infinite | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_quaternion : #ℍ[R] = #R ^ 4 | mk_quaternion_algebra _ _ | lemma | cardinal.mk_quaternion | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | The cardinality of the quaternions, as a type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_quaternion_of_infinite [infinite R] : #ℍ[R] = #R | by rw [mk_quaternion, pow_four] | lemma | cardinal.mk_quaternion_of_infinite | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_univ_quaternion : #(set.univ : set ℍ[R]) = #R ^ 4 | mk_univ_quaternion_algebra _ _ | lemma | cardinal.mk_univ_quaternion | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | The cardinality of the quaternions, as a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_univ_quaternion_of_infinite [infinite R] : #(set.univ : set ℍ[R]) = #R | by rw [mk_univ_quaternion, pow_four] | lemma | cardinal.mk_univ_quaternion_of_infinite | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis {R : Type*} (A : Type*) [comm_ring R] [ring A] [algebra R A] (c₁ c₂ : R) | (i j k : A)
(i_mul_i : i * i = c₁ • 1)
(j_mul_j : j * j = c₂ • 1)
(i_mul_j : i * j = k)
(j_mul_i : j * i = -k) | structure | quaternion_algebra.basis | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"algebra",
"basis",
"comm_ring",
"ring"
] | A quaternion basis contains the information both sufficient and necessary to construct an
`R`-algebra homomorphism from `ℍ[R,c₁,c₂]` to `A`; or equivalently, a surjective
`R`-algebra homomorphism from `ℍ[R,c₁,c₂]` to an `R`-subalgebra of `A`.
Note that for definitional convenience, `k` is provided as a field even thou... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext ⦃q₁ q₂ : basis A c₁ c₂⦄ (hi : q₁.i = q₂.i) (hj : q₁.j = q₂.j) :
q₁ = q₂ | begin
cases q₁,
cases q₂,
congr',
rw [←q₁_i_mul_j, ←q₂_i_mul_j],
congr'
end | lemma | quaternion_algebra.basis.ext | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"basis"
] | Since `k` is redundant, it is not necessary to show `q₁.k = q₂.k` when showing `q₁ = q₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self : basis ℍ[R,c₁,c₂] c₁ c₂ | { i := ⟨0, 1, 0, 0⟩,
i_mul_i := by { ext; simp },
j := ⟨0, 0, 1, 0⟩,
j_mul_j := by { ext; simp },
k := ⟨0, 0, 0, 1⟩,
i_mul_j := by { ext; simp },
j_mul_i := by { ext; simp } } | def | quaternion_algebra.basis.self | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"basis"
] | There is a natural quaternionic basis for the `quaternion_algebra`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
i_mul_k : q.i * q.k = c₁ • q.j | by rw [←i_mul_j, ←mul_assoc, i_mul_i, smul_mul_assoc, one_mul] | lemma | quaternion_algebra.basis.i_mul_k | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"one_mul",
"smul_mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
k_mul_i : q.k * q.i = -c₁ • q.j | by rw [←i_mul_j, mul_assoc, j_mul_i, mul_neg, i_mul_k, neg_smul] | lemma | quaternion_algebra.basis.k_mul_i | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"mul_assoc",
"mul_neg",
"neg_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
k_mul_j : q.k * q.j = c₂ • q.i | by rw [←i_mul_j, mul_assoc, j_mul_j, mul_smul_comm, mul_one] | lemma | quaternion_algebra.basis.k_mul_j | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"mul_assoc",
"mul_one",
"mul_smul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
j_mul_k : q.j * q.k = -c₂ • q.i | by rw [←i_mul_j, ←mul_assoc, j_mul_i, neg_mul, k_mul_j, neg_smul] | lemma | quaternion_algebra.basis.j_mul_k | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"neg_mul",
"neg_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
k_mul_k : q.k * q.k = -((c₁ * c₂) • 1) | by rw [←i_mul_j, mul_assoc, ←mul_assoc q.j _ _, j_mul_i, ←i_mul_j,
←mul_assoc, mul_neg, ←mul_assoc, i_mul_i, smul_mul_assoc, one_mul,
neg_mul, smul_mul_assoc, j_mul_j, smul_smul] | lemma | quaternion_algebra.basis.k_mul_k | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"mul_assoc",
"mul_neg",
"neg_mul",
"one_mul",
"smul_mul_assoc",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (x : ℍ[R,c₁,c₂]) : A | algebra_map R _ x.re + x.im_i • q.i + x.im_j • q.j + x.im_k • q.k | def | quaternion_algebra.basis.lift | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"algebra_map",
"lift"
] | Intermediate result used to define `quaternion_algebra.basis.lift_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_zero : q.lift (0 : ℍ[R,c₁,c₂]) = 0 | by simp [lift] | lemma | quaternion_algebra.basis.lift_zero | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_one : q.lift (1 : ℍ[R,c₁,c₂]) = 1 | by simp [lift] | lemma | quaternion_algebra.basis.lift_one | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_add (x y : ℍ[R,c₁,c₂]) : q.lift (x + y) = q.lift x + q.lift y | by { simp [lift, add_smul], abel } | lemma | quaternion_algebra.basis.lift_add | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"add_smul",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mul (x y : ℍ[R,c₁,c₂]) : q.lift (x * y) = q.lift x * q.lift y | begin
simp only [lift, algebra.algebra_map_eq_smul_one],
simp only [add_mul],
simp only [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one,
←algebra.smul_def, smul_add, smul_smul],
simp only [i_mul_i, j_mul_j, i_mul_j, j_mul_i, i_mul_k, k_mul_i, k_mul_j, j_mul_k, k_mul_k],
simp only [smul_... | lemma | quaternion_algebra.basis.lift_mul | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"add_smul",
"algebra.algebra_map_eq_smul_one",
"lift",
"mul_comm",
"mul_neg",
"mul_one",
"mul_right_comm",
"mul_smul_comm",
"neg_smul",
"one_mul",
"smul_add",
"smul_mul_assoc",
"smul_neg",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_smul (r : R) (x : ℍ[R,c₁,c₂]) : q.lift (r • x) = r • q.lift x | by simp [lift, mul_smul, ←algebra.smul_def] | lemma | quaternion_algebra.basis.lift_smul | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom : ℍ[R,c₁,c₂] →ₐ[R] A | alg_hom.mk'
{ to_fun := q.lift,
map_zero' := q.lift_zero,
map_one' := q.lift_one,
map_add' := q.lift_add,
map_mul' := q.lift_mul }
q.lift_smul | def | quaternion_algebra.basis.lift_hom | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"alg_hom.mk'"
] | A `quaternion_algebra.basis` implies an `alg_hom` from the quaternions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_hom (F : A →ₐ[R] B) : basis B c₁ c₂ | { i := F q.i,
i_mul_i := by rw [←F.map_mul, q.i_mul_i, F.map_smul, F.map_one],
j := F q.j,
j_mul_j := by rw [←F.map_mul, q.j_mul_j, F.map_smul, F.map_one],
k := F q.k,
i_mul_j := by rw [←F.map_mul, q.i_mul_j],
j_mul_i := by rw [←F.map_mul, q.j_mul_i, F.map_neg], } | def | quaternion_algebra.basis.comp_hom | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"basis"
] | Transform a `quaternion_algebra.basis` through an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : basis A c₁ c₂ ≃ (ℍ[R,c₁,c₂] →ₐ[R] A) | { to_fun := basis.lift_hom,
inv_fun := (basis.self R).comp_hom,
left_inv := λ q, begin
ext;
simp [basis.lift],
end,
right_inv := λ F, begin
ext,
dsimp [basis.lift],
rw ←F.commutes,
simp only [←F.commutes, ←F.map_smul, ←F.map_add, mk_add_mk, smul_mk, smul_zero, algebra_map_eq],
congr,... | def | quaternion_algebra.lift | algebra | src/algebra/quaternion_basis.lean | [
"algebra.quaternion",
"tactic.ring"
] | [
"basis",
"inv_fun",
"lift",
"smul_zero"
] | A quaternionic basis on `A` is equivalent to a map from the quaternion algebra to `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_quotient (A : out_param $ Type u) (B : Type v) | (quotient' : B → Type (max u v)) | class | has_quotient | algebra | src/algebra/quotient.lean | [
"tactic.basic"
] | [] | `has_quotient A B` is a notation typeclass that allows us to write `A ⧸ b` for `b : B`.
This allows the usual notation for quotients of algebraic structures,
such as groups, modules and rings.
`A` is a parameter, despite being unused in the definition below, so it appears in the notation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_quotient.quotient (A : out_param $ Type u) {B : Type v} [has_quotient A B] (b : B) :
Type (max u v) | has_quotient.quotient' b | def | has_quotient.quotient | algebra | src/algebra/quotient.lean | [
"tactic.basic"
] | [
"has_quotient"
] | `has_quotient.quotient A b` (with notation `A ⧸ b`) is the quotient of the type `A` by `b`.
This differs from `has_quotient.quotient'` in that the `A` argument is explicit, which is necessary
to make Lean show the notation in the goal state. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_algebra_map (c : ring_con A) (s : S) :
(↑(algebra_map S A s) : c.quotient) = algebra_map S _ s | rfl | lemma | ring_con.coe_algebra_map | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"ring_con"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel (r : R → R → Prop) : R → R → Prop
| of ⦃x y : R⦄ (h : r x y) : rel x y
| add_left ⦃a b c⦄ : rel a b → rel (a + c) (b + c)
| mul_left ⦃a b c⦄ : rel a b → rel (a * c) (b * c)
| mul_right ⦃a b c⦄ : rel b c → rel (a * b) (a * c) | inductive | ring_quot.rel | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel"
] | Given an arbitrary relation `r` on a ring, we strengthen it to a relation `rel r`,
such that the equivalence relation generated by `rel r` has `x ~ y` if and only if
`x - y` is in the ideal generated by elements `a - b` such that `r a b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.add_right {r : R → R → Prop} ⦃a b c : R⦄ (h : rel r b c) : rel r (a + b) (a + c) | by { rw [add_comm a b, add_comm a c], exact rel.add_left h } | theorem | ring_quot.rel.add_right | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.neg {R : Type u₁} [ring R] {r : R → R → Prop} ⦃a b : R⦄ (h : rel r a b) :
rel r (-a) (-b) | by simp only [neg_eq_neg_one_mul a, neg_eq_neg_one_mul b, rel.mul_right h] | theorem | ring_quot.rel.neg | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"neg_eq_neg_one_mul",
"rel",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.sub_left {R : Type u₁} [ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : rel r a b) :
rel r (a - c) (b - c) | by simp only [sub_eq_add_neg, h.add_left] | theorem | ring_quot.rel.sub_left | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.sub_right {R : Type u₁} [ring R] {r : R → R → Prop} ⦃a b c : R⦄ (h : rel r b c) :
rel r (a - b) (a - c) | by simp only [sub_eq_add_neg, h.neg.add_right] | theorem | ring_quot.rel.sub_right | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel.smul {r : A → A → Prop} (k : S) ⦃a b : A⦄ (h : rel r a b) : rel r (k • a) (k • b) | by simp only [algebra.smul_def, rel.mul_right h] | theorem | ring_quot.rel.smul | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"algebra.smul_def",
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_con (r : R → R → Prop) : ring_con R | { r := eqv_gen (rel r),
iseqv := eqv_gen.is_equivalence _,
add' := λ a b c d hab hcd, begin
induction hab with a' b' hab e a' b' hab' _ c' d' e hcd' hde' ihcd' ihde' generalizing c d,
{ refine (eqv_gen.rel _ _ hab.add_left).trans _ _ _ _,
induction hcd with c' d' hcd f c' d' hcd' habcd' c' d' f' hcd' ... | def | ring_quot.ring_con | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel",
"ring_con"
] | `eqv_gen (ring_quot.rel r)` is a ring congruence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eqv_gen_rel_eq (r : R → R → Prop) : eqv_gen (rel r) = ring_con_gen.rel r | begin
ext x₁ x₂,
split,
{ intro h,
induction h with x₃ x₄ h₃₄,
{ induction h₃₄ with _ dfg h₃₄ x₃ x₄ x₅ h₃₄',
{ exact ring_con_gen.rel.of _ _ ‹_› },
{ exact h₃₄_ih.add (ring_con_gen.rel.refl _) },
{ exact h₃₄_ih.mul (ring_con_gen.rel.refl _) },
{ exact (ring_con_gen.rel.refl _).mul ... | lemma | ring_quot.eqv_gen_rel_eq | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel",
"ring_con_gen.rel",
"ring_quot.ring_con"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_quot (r : R → R → Prop) | (to_quot : quot (ring_quot.rel r)) | structure | ring_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot.rel"
] | The quotient of a ring by an arbitrary relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_cast (n : ℕ) : ring_quot r | ⟨quot.mk _ n⟩ | def | ring_quot.nat_cast | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero : ring_quot r | ⟨quot.mk _ 0⟩ | def | ring_quot.zero | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one : ring_quot r | ⟨quot.mk _ 1⟩ | def | ring_quot.one | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add : ring_quot r → ring_quot r → ring_quot r | | ⟨a⟩ ⟨b⟩ := ⟨quot.map₂ (+) rel.add_right rel.add_left a b⟩ | def | ring_quot.add | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul : ring_quot r → ring_quot r → ring_quot r | | ⟨a⟩ ⟨b⟩ := ⟨quot.map₂ (*) rel.mul_right rel.mul_left a b⟩ | def | ring_quot.mul | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg {R : Type u₁} [ring R] (r : R → R → Prop) : ring_quot r → ring_quot r | | ⟨a⟩:= ⟨quot.map (λ a, -a) rel.neg a⟩ | def | ring_quot.neg | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub {R : Type u₁} [ring R] (r : R → R → Prop) :
ring_quot r → ring_quot r → ring_quot r | | ⟨a⟩ ⟨b⟩ := ⟨quot.map₂ has_sub.sub rel.sub_right rel.sub_left a b⟩ | def | ring_quot.sub | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
npow (n : ℕ) : ring_quot r → ring_quot r | | ⟨a⟩ := ⟨quot.lift
(λ a, quot.mk (ring_quot.rel r) (a ^ n))
(λ a b (h : rel r a b), begin
-- note we can't define a `rel.pow` as `rel` isn't reflexive so `rel r 1 1` isn't true
dsimp only,
induction n,
{ rw [pow_zero, pow_zero] },
{ rw [po... | def | ring_quot.npow | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"congr_arg2",
"pow_succ",
"pow_zero",
"rel",
"ring_quot",
"ring_quot.rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul [algebra S R] (n : S) : ring_quot r → ring_quot r | | ⟨a⟩ := ⟨quot.map (λ a, n • a) (rel.smul n) a⟩ | def | ring_quot.smul | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"algebra",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_quot : (⟨quot.mk _ 0⟩ : ring_quot r) = 0 | show _ = zero r, by rw zero | lemma | ring_quot.zero_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_quot : (⟨quot.mk _ 1⟩ : ring_quot r) = 1 | show _ = one r, by rw one | lemma | ring_quot.one_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_quot {a b} : (⟨quot.mk _ a⟩ + ⟨quot.mk _ b⟩ : ring_quot r) = ⟨quot.mk _ (a + b)⟩ | by { show add r _ _ = _, rw add, refl } | lemma | ring_quot.add_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_quot {a b} : (⟨quot.mk _ a⟩ * ⟨quot.mk _ b⟩ : ring_quot r) = ⟨quot.mk _ (a * b)⟩ | by { show mul r _ _ = _, rw mul, refl } | lemma | ring_quot.mul_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_quot {a} {n : ℕ}: (⟨quot.mk _ a⟩ ^ n : ring_quot r) = ⟨quot.mk _ (a ^ n)⟩ | by { show npow r _ _ = _, rw npow } | lemma | ring_quot.pow_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_quot {R : Type u₁} [ring R] (r : R → R → Prop) {a} :
(-⟨quot.mk _ a⟩ : ring_quot r) = ⟨quot.mk _ (-a)⟩ | by { show neg r _ = _, rw neg, refl } | lemma | ring_quot.neg_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_quot {R : Type u₁} [ring R] (r : R → R → Prop) {a b} :
(⟨quot.mk _ a⟩ - ⟨ quot.mk _ b⟩ : ring_quot r) = ⟨quot.mk _ (a - b)⟩ | by { show sub r _ _ = _, rw sub, refl } | lemma | ring_quot.sub_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_quot [algebra S R] {n : S} {a : R} :
(n • ⟨quot.mk _ a⟩ : ring_quot r) = ⟨quot.mk _ (n • a)⟩ | by { show smul r _ _ = _, rw smul, refl } | lemma | ring_quot.smul_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"algebra",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ring_hom (r : R → R → Prop) : R →+* ring_quot r | { to_fun := λ x, ⟨quot.mk _ x⟩,
map_one' := by simp [← one_quot],
map_mul' := by simp [mul_quot],
map_zero' := by simp [← zero_quot],
map_add' := by simp [add_quot], } | def | ring_quot.mk_ring_hom | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | The quotient map from a ring to its quotient, as a homomorphism of rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_ring_hom_rel {r : R → R → Prop} {x y : R} (w : r x y) :
mk_ring_hom r x = mk_ring_hom r y | by simp [mk_ring_hom, quot.sound (rel.of w)] | lemma | ring_quot.mk_ring_hom_rel | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ring_hom_surjective (r : R → R → Prop) : function.surjective (mk_ring_hom r) | by { dsimp [mk_ring_hom], rintro ⟨⟨⟩⟩, simp, } | lemma | ring_quot.mk_ring_hom_surjective | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_quot_ext {T : Type u₄} [semiring T] {r : R → R → Prop} (f g : ring_quot r →+* T)
(w : f.comp (mk_ring_hom r) = g.comp (mk_ring_hom r)) : f = g | begin
ext,
rcases mk_ring_hom_surjective r x with ⟨x, rfl⟩,
exact (ring_hom.congr_fun w x : _),
end | lemma | ring_quot.ring_quot_ext | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_hom.congr_fun",
"ring_quot",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift {r : R → R → Prop} :
{f : R →+* T // ∀ ⦃x y⦄, r x y → f x = f y} ≃ (ring_quot r →+* T) | { to_fun := λ f', let f := (f' : R →+* T) in
{ to_fun := λ x, quot.lift f
begin
rintros _ _ r,
induction r,
case of : _ _ r { exact f'.prop r, },
case add_left : _ _ _ _ r' { simp [r'], },
case mul_left : _ _ _ _ r' { simp [r'], },
case mul_right : _ _ _ _ r' { simp [r'], }, | def | ring_quot.lift | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift",
"ring_quot"
] | Any ring homomorphism `f : R →+* T` which respects a relation `r : R → R → Prop`
factors uniquely through a morphism `ring_quot r →+* T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk_ring_hom_apply (f : R →+* T) {r : R → R → Prop} (w : ∀ ⦃x y⦄, r x y → f x = f y) (x) :
lift ⟨f, w⟩ (mk_ring_hom r x) = f x | by { simp_rw [lift, mk_ring_hom], refl } | lemma | lift_mk_ring_hom_apply | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (f : R →+* T) {r : R → R → Prop} (w : ∀ ⦃x y⦄, r x y → f x = f y)
(g : ring_quot r →+* T) (h : g.comp (mk_ring_hom r) = f) : g = lift ⟨f, w⟩ | by { ext, simp [h], } | lemma | lift_unique | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lift_comp_mk_ring_hom {r : R → R → Prop} (f : ring_quot r →+* T) :
f = lift ⟨f.comp (mk_ring_hom r), λ x y h, by { dsimp, rw mk_ring_hom_rel h, }⟩ | begin
conv_lhs { rw ← lift.apply_symm_apply f },
rw lift,
refl,
end | lemma | eq_lift_comp_mk_ring_hom | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_quot_to_ideal_quotient (r : B → B → Prop) :
ring_quot r →+* B ⧸ ideal.of_rel r | lift
⟨ideal.quotient.mk (ideal.of_rel r),
λ x y h, ideal.quotient.eq.2 $ submodule.mem_Inf.mpr (λ p w, w ⟨x, y, h, sub_add_cancel x y⟩)⟩ | def | ring_quot_to_ideal_quotient | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ideal.of_rel",
"lift",
"ring_quot"
] | The universal ring homomorphism from `ring_quot r` to `B ⧸ ideal.of_rel r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_quot_to_ideal_quotient_apply (r : B → B → Prop) (x : B) :
ring_quot_to_ideal_quotient r (mk_ring_hom r x) = ideal.quotient.mk _ x | begin
simp_rw [ring_quot_to_ideal_quotient, lift, mk_ring_hom],
refl
end | lemma | ring_quot_to_ideal_quotient_apply | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"lift",
"ring_quot_to_ideal_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_quotient_to_ring_quot (r : B → B → Prop) :
B ⧸ ideal.of_rel r →+* ring_quot r | ideal.quotient.lift (ideal.of_rel r) (mk_ring_hom r)
begin
refine λ x h, submodule.span_induction h _ _ _ _,
{ rintro y ⟨a, b, h, su⟩,
symmetry' at su,
rw ←sub_eq_iff_eq_add at su,
rw [ ← su, ring_hom.map_sub, mk_ring_hom_rel h, sub_self], },
{ simp, },
{ intros a b ha hb, simp [ha, hb], },
{ intr... | def | ideal_quotient_to_ring_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ideal.of_rel",
"ideal.quotient.lift",
"ring_hom.map_sub",
"ring_quot",
"submodule.span_induction"
] | The universal ring homomorphism from `B ⧸ ideal.of_rel r` to `ring_quot r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_quotient_to_ring_quot_apply (r : B → B → Prop) (x : B) :
ideal_quotient_to_ring_quot r (ideal.quotient.mk _ x) = mk_ring_hom r x | rfl | lemma | ideal_quotient_to_ring_quot_apply | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"ideal_quotient_to_ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_quot_equiv_ideal_quotient (r : B → B → Prop) :
ring_quot r ≃+* B ⧸ ideal.of_rel r | ring_equiv.of_hom_inv (ring_quot_to_ideal_quotient r) (ideal_quotient_to_ring_quot r)
(begin
ext,
simp_rw [ring_quot_to_ideal_quotient, lift, mk_ring_hom],
dsimp,
rw [mk_ring_hom],
refl
end)
(begin
ext,
simp_rw [ring_quot_to_ideal_quotient, lift, mk_ring_hom],
dsimp,
rw [mk_rin... | def | ring_quot_equiv_ideal_quotient | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ideal.of_rel",
"ideal_quotient_to_ring_quot",
"lift",
"ring_equiv.of_hom_inv",
"ring_quot",
"ring_quot_to_ideal_quotient"
] | The ring equivalence between `ring_quot r` and `(ideal.of_rel r).quotient` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel.star ⦃a b : R⦄ (h : rel r a b) :
rel r (star a) (star b) | begin
induction h,
{ exact rel.of (hr _ _ h_h) },
{ rw [star_add, star_add], exact rel.add_left h_ih, },
{ rw [star_mul, star_mul], exact rel.mul_right h_ih, },
{ rw [star_mul, star_mul], exact rel.mul_left h_ih, },
end | theorem | rel.star | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star' : ring_quot r → ring_quot r | | ⟨a⟩ := ⟨quot.map (star : R → R) (rel.star r hr) a⟩ | def | star' | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"rel.star",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star'_quot (hr : ∀ a b, r a b → r (star a) (star b)) {a} :
(star' r hr ⟨quot.mk _ a⟩ : ring_quot r) = ⟨quot.mk _ (star a)⟩ | by { show star' r _ _ = _, rw star', refl } | lemma | star'_quot | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot",
"star'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_ring {R : Type u₁} [semiring R] [star_ring R] (r : R → R → Prop)
(hr : ∀ a b, r a b → r (star a) (star b)) :
star_ring (ring_quot r) | { star := star' r hr,
star_involutive := by { rintros ⟨⟨⟩⟩, simp [star'_quot], },
star_mul := by { rintros ⟨⟨⟩⟩ ⟨⟨⟩⟩, simp [star'_quot, mul_quot, star_mul], },
star_add := by { rintros ⟨⟨⟩⟩ ⟨⟨⟩⟩, simp [star'_quot, add_quot, star_add], } } | def | star_ring | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot",
"semiring",
"star'",
"star'_quot"
] | Transfer a star_ring instance through a quotient, if the quotient is invariant to `star` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_alg_hom (s : A → A → Prop) : A →ₐ[S] ring_quot s | { commutes' := λ r, by { simp [mk_ring_hom], refl },
..mk_ring_hom s } | def | mk_alg_hom | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | The quotient map from an `S`-algebra to its quotient, as a homomorphism of `S`-algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_alg_hom_coe (s : A → A → Prop) : (mk_alg_hom S s : A →+* ring_quot s) = mk_ring_hom s | by { simp_rw [mk_alg_hom, mk_ring_hom], refl } | lemma | mk_alg_hom_coe | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"mk_alg_hom",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_alg_hom_rel {s : A → A → Prop} {x y : A} (w : s x y) :
mk_alg_hom S s x = mk_alg_hom S s y | by simp [mk_alg_hom, mk_ring_hom, quot.sound (rel.of w)] | lemma | mk_alg_hom_rel | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"mk_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_alg_hom_surjective (s : A → A → Prop) : function.surjective (mk_alg_hom S s) | by { dsimp [mk_alg_hom, mk_ring_hom], rintro ⟨⟨a⟩⟩, use a, refl, } | lemma | mk_alg_hom_surjective | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"mk_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_quot_ext' {s : A → A → Prop}
(f g : ring_quot s →ₐ[S] B) (w : f.comp (mk_alg_hom S s) = g.comp (mk_alg_hom S s)) : f = g | begin
ext,
rcases mk_alg_hom_surjective S s x with ⟨x, rfl⟩,
exact (alg_hom.congr_fun w x : _),
end | lemma | ring_quot_ext' | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"alg_hom.congr_fun",
"mk_alg_hom",
"mk_alg_hom_surjective",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_alg_hom {s : A → A → Prop} :
{f : A →ₐ[S] B // ∀ ⦃x y⦄, s x y → f x = f y} ≃ (ring_quot s →ₐ[S] B) | { to_fun := λ f', let f := (f' : A →ₐ[S] B) in
{ to_fun := λ x, quot.lift f
begin
rintros _ _ r,
induction r,
case of : _ _ r { exact f'.prop r, },
case add_left : _ _ _ _ r' { simp [r'], },
case mul_left : _ _ _ _ r' { simp [r'], },
case mul_right : _ _ _ _ r' { simp [r'], }, | def | lift_alg_hom | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"ring_quot"
] | Any `S`-algebra homomorphism `f : A →ₐ[S] B` which respects a relation `s : A → A → Prop`
factors uniquely through a morphism `ring_quot s →ₐ[S] B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_alg_hom_mk_alg_hom_apply (f : A →ₐ[S] B) {s : A → A → Prop}
(w : ∀ ⦃x y⦄, s x y → f x = f y) (x) :
(lift_alg_hom S ⟨f, w⟩) ((mk_alg_hom S s) x) = f x | by { simp_rw [lift_alg_hom, mk_alg_hom, mk_ring_hom], refl, } | lemma | lift_alg_hom_mk_alg_hom_apply | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift_alg_hom",
"mk_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_alg_hom_unique (f : A →ₐ[S] B) {s : A → A → Prop} (w : ∀ ⦃x y⦄, s x y → f x = f y)
(g : ring_quot s →ₐ[S] B) (h : g.comp (mk_alg_hom S s) = f) : g = lift_alg_hom S ⟨f, w⟩ | by { ext, simp [h], } | lemma | lift_alg_hom_unique | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift_alg_hom",
"mk_alg_hom",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lift_alg_hom_comp_mk_alg_hom {s : A → A → Prop} (f : ring_quot s →ₐ[S] B) :
f = lift_alg_hom S ⟨f.comp (mk_alg_hom S s), λ x y h, by { dsimp, erw mk_alg_hom_rel S h, }⟩ | begin
conv_lhs { rw ← ((lift_alg_hom S).apply_symm_apply f) },
rw lift_alg_hom,
refl,
end | lemma | eq_lift_alg_hom_comp_mk_alg_hom | algebra | src/algebra/ring_quot.lean | [
"algebra.algebra.hom",
"ring_theory.ideal.quotient"
] | [
"lift_alg_hom",
"mk_alg_hom",
"mk_alg_hom_rel",
"ring_quot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_with_zero [has_zero R] [has_zero M] extends smul_zero_class R M | (zero_smul : ∀ m : M, (0 : R) • m = 0) | class | smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_zero_class",
"zero_smul"
] | `smul_with_zero` is a class consisting of a Type `R` with `0 ∈ R` and a scalar multiplication
of `R` on a Type `M` with `0`, such that the equality `r • m = 0` holds if at least one among `r`
or `m` equals `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_zero_class.to_smul_with_zero [mul_zero_class R] : smul_with_zero R R | { smul := (*),
smul_zero := mul_zero,
zero_smul := zero_mul } | instance | mul_zero_class.to_smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_zero",
"mul_zero_class",
"smul_with_zero",
"smul_zero",
"zero_mul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_class.to_opposite_smul_with_zero [mul_zero_class R] : smul_with_zero Rᵐᵒᵖ R | { smul := (•),
smul_zero := λ r, zero_mul _,
zero_smul := mul_zero } | instance | mul_zero_class.to_opposite_smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_zero",
"mul_zero_class",
"smul_with_zero",
"smul_zero",
"zero_mul",
"zero_smul"
] | Like `mul_zero_class.to_smul_with_zero`, but multiplies on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_smul (m : M) : (0 : R) • m = 0 | smul_with_zero.zero_smul m | lemma | zero_smul | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_zero_of_left (h : a = 0) (b : M) : a • b = 0 | h.symm ▸ zero_smul _ b | lemma | smul_eq_zero_of_left | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_zero_of_right (a : R) (h : b = 0) : a • b = 0 | h.symm ▸ smul_zero a | lemma | smul_eq_zero_of_right | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_ne_zero_of_smul : a • b ≠ 0 → a ≠ 0 | mt $ λ h, smul_eq_zero_of_left h b | lemma | left_ne_zero_of_smul | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_eq_zero_of_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_ne_zero_of_smul : a • b ≠ 0 → b ≠ 0 | mt $ smul_eq_zero_of_right a | lemma | right_ne_zero_of_smul | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_eq_zero_of_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.smul_with_zero
(f : zero_hom M' M) (hf : function.injective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
smul_with_zero R M' | { smul := (•),
zero_smul := λ a, hf $ by simp [smul],
smul_zero := λ a, hf $ by simp [smul]} | def | function.injective.smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_with_zero",
"smul_zero",
"zero_hom",
"zero_smul"
] | Pullback a `smul_with_zero` structure along an injective zero-preserving homomorphism.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.smul_with_zero
(f : zero_hom M M') (hf : function.surjective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
smul_with_zero R M' | { smul := (•),
zero_smul := λ m, by { rcases hf m with ⟨x, rfl⟩, simp [←smul] },
smul_zero := λ c, by simp only [← f.map_zero, ← smul, smul_zero] } | def | function.surjective.smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_with_zero",
"smul_zero",
"zero_hom",
"zero_smul"
] | Pushforward a `smul_with_zero` structure along a surjective zero-preserving homomorphism.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_with_zero.comp_hom (f : zero_hom R' R) : smul_with_zero R' M | { smul := (•) ∘ f,
smul_zero := λ m, by simp,
zero_smul := λ m, by simp } | def | smul_with_zero.comp_hom | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"smul_with_zero",
"smul_zero",
"zero_hom",
"zero_smul"
] | Compose a `smul_with_zero` with a `zero_hom`, with action `f r' • m` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid.nat_smul_with_zero [add_monoid M] : smul_with_zero ℕ M | { smul_zero := nsmul_zero,
zero_smul := zero_nsmul } | instance | add_monoid.nat_smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"add_monoid",
"smul_with_zero",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group.int_smul_with_zero [add_group M] : smul_with_zero ℤ M | { smul_zero := zsmul_zero,
zero_smul := zero_zsmul } | instance | add_group.int_smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"add_group",
"smul_with_zero",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero extends mul_action R M | -- these fields are copied from `smul_with_zero`, as `extends` behaves poorly
(smul_zero : ∀ r : R, r • (0 : M) = 0)
(zero_smul : ∀ m : M, (0 : R) • m = 0) | class | mul_action_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action",
"smul_zero",
"zero_smul"
] | An action of a monoid with zero `R` on a Type `M`, also with `0`, extends `mul_action` and
is compatible with `0` (both in `R` and in `M`), with `1 ∈ R`, and with associativity of
multiplication on the monoid `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_action_with_zero.to_smul_with_zero [m : mul_action_with_zero R M] :
smul_with_zero R M | {..m} | instance | mul_action_with_zero.to_smul_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action_with_zero",
"smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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