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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|---|---|---|---|---|---|---|---|---|---|---|
mul_support_mul_single [decidable_eq B] :
mul_support (mul_single a b) = if b = 1 then ∅ else {a} | by { split_ifs with h; simp [h] } | lemma | pi.mul_support_mul_single | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_single_disjoint {b' : B} (hb : b ≠ 1) (hb' : b' ≠ 1) {i j : A} :
disjoint (mul_support (mul_single i b)) (mul_support (mul_single j b')) ↔ i ≠ j | by rw [mul_support_mul_single_of_ne hb, mul_support_mul_single_of_ne hb', disjoint_singleton] | lemma | pi.mul_support_mul_single_disjoint | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_alg (α : Type*) : Type* | α | def | sym_alg | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | The symmetrized algebra has the same underlying space as the original algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sym : α ≃ αˢʸᵐ | equiv.refl _ | def | sym_alg.sym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"equiv.refl",
"sym"
] | The element of `sym_alg α` that represents `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unsym : αˢʸᵐ ≃ α | equiv.refl _ | def | sym_alg.unsym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"equiv.refl"
] | The element of `α` represented by `x : αˢʸᵐ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unsym_sym (a : α) : unsym (sym a) = a | rfl | lemma | sym_alg.unsym_sym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_unsym (a : α) : sym (unsym a) = a | rfl | lemma | sym_alg.sym_unsym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_comp_unsym : (sym : α → αˢʸᵐ) ∘ unsym = id | rfl | lemma | sym_alg.sym_comp_unsym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_comp_sym : (unsym : αˢʸᵐ → α) ∘ sym = id | rfl | lemma | sym_alg.unsym_comp_sym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_symm : (@sym α).symm = unsym | rfl | lemma | sym_alg.sym_symm | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_symm : (@unsym α).symm = sym | rfl | lemma | sym_alg.unsym_symm | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_bijective : bijective (sym : α → αˢʸᵐ) | sym.bijective | lemma | sym_alg.sym_bijective | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_bijective : bijective (unsym : αˢʸᵐ → α) | unsym.symm.bijective | lemma | sym_alg.unsym_bijective | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_injective : injective (sym : α → αˢʸᵐ) | sym.injective | lemma | sym_alg.sym_injective | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_surjective : surjective (sym : α → αˢʸᵐ) | sym.surjective | lemma | sym_alg.sym_surjective | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_injective : injective (unsym : αˢʸᵐ → α) | unsym.injective | lemma | sym_alg.unsym_injective | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_surjective : surjective (unsym : αˢʸᵐ → α) | unsym.surjective | lemma | sym_alg.unsym_surjective | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_inj {a b : α} : sym a = sym b ↔ a = b | sym_injective.eq_iff | lemma | sym_alg.sym_inj | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_inj {a b : αˢʸᵐ} : unsym a = unsym b ↔ a = b | unsym_injective.eq_iff | lemma | sym_alg.unsym_inj | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_one [has_one α] : sym (1 : α) = 1 | rfl | lemma | sym_alg.sym_one | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_one [has_one α] : unsym (1 : αˢʸᵐ) = 1 | rfl | lemma | sym_alg.unsym_one | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_add [has_add α] (a b : α) : sym (a + b) = sym a + sym b | rfl | lemma | sym_alg.sym_add | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_add [has_add α] (a b : αˢʸᵐ) : unsym (a + b) = unsym a + unsym b | rfl | lemma | sym_alg.unsym_add | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_sub [has_sub α] (a b : α) : sym (a - b) = sym a - sym b | rfl | lemma | sym_alg.sym_sub | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_sub [has_sub α] (a b : αˢʸᵐ) : unsym (a - b) = unsym a - unsym b | rfl | lemma | sym_alg.unsym_sub | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_neg [has_neg α] (a : α) : sym (-a) = -sym a | rfl | lemma | sym_alg.sym_neg | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_neg [has_neg α] (a : αˢʸᵐ) : unsym (-a) = -unsym a | rfl | lemma | sym_alg.unsym_neg | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def [has_add α] [has_mul α] [has_one α] [invertible (2 : α)] (a b : αˢʸᵐ) :
a * b = sym (⅟2*(unsym a * unsym b + unsym b * unsym a)) | by refl | lemma | sym_alg.mul_def | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"invertible",
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_mul [has_mul α] [has_add α] [has_one α] [invertible (2 : α)] (a b : αˢʸᵐ) :
unsym (a * b) = ⅟2*(unsym a * unsym b + unsym b * unsym a) | by refl | lemma | sym_alg.unsym_mul | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"invertible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_mul_sym [has_mul α] [has_add α] [has_one α] [invertible (2 : α)] (a b : α) :
sym a * sym b = sym (⅟2*(a * b + b * a)) | rfl | lemma | sym_alg.sym_mul_sym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"invertible",
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_inv [has_inv α] (a : α) : sym (a⁻¹) = (sym a)⁻¹ | rfl | lemma | sym_alg.sym_inv | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_inv [has_inv α] (a : αˢʸᵐ) : unsym (a⁻¹) = (unsym a)⁻¹ | rfl | lemma | sym_alg.unsym_inv | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_smul {R : Type*} [has_smul R α] (c : R) (a : α) : sym (c • a) = c • sym a | rfl | lemma | sym_alg.sym_smul | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"has_smul",
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_smul {R : Type*} [has_smul R α] (c : R) (a : αˢʸᵐ) :
unsym (c • a) = c • unsym a | rfl | lemma | sym_alg.unsym_smul | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_eq_one_iff [has_one α] (a : αˢʸᵐ) : a.unsym = 1 ↔ a = 1 | unsym_injective.eq_iff' rfl | lemma | sym_alg.unsym_eq_one_iff | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_eq_one_iff [has_one α] (a : α) : sym a = 1 ↔ a = 1 | sym_injective.eq_iff' rfl | lemma | sym_alg.sym_eq_one_iff | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_ne_one_iff [has_one α] (a : αˢʸᵐ) : a.unsym ≠ (1 : α) ↔ a ≠ (1 : αˢʸᵐ) | not_congr $ unsym_eq_one_iff a | lemma | sym_alg.unsym_ne_one_iff | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_ne_one_iff [has_one α] (a : α) : sym a ≠ (1 : αˢʸᵐ) ↔ a ≠ (1 : α) | not_congr $ sym_eq_one_iff a | lemma | sym_alg.sym_ne_one_iff | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_sym [has_mul α] [has_add α] [has_one α] [invertible (2 : α)] (a : α)
[invertible a] : ⅟(sym a) = sym (⅟a) | rfl | lemma | sym_alg.inv_of_sym | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"invertible",
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unsym_mul_self [semiring α] [invertible (2 : α)] (a : αˢʸᵐ) :
unsym (a*a) = unsym a * unsym a | by rw [mul_def, unsym_sym, ←two_mul, inv_of_mul_self_assoc] | lemma | sym_alg.unsym_mul_self | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"inv_of_mul_self_assoc",
"invertible",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sym_mul_self [semiring α] [invertible (2 : α)] (a : α) : sym (a*a) = sym a * sym a | by rw [sym_mul_sym, ←two_mul, inv_of_mul_self_assoc] | lemma | sym_alg.sym_mul_self | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"inv_of_mul_self_assoc",
"invertible",
"semiring",
"sym"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm [has_mul α] [add_comm_semigroup α] [has_one α] [invertible (2 : α)] (a b : αˢʸᵐ) :
a * b = b * a | by rw [mul_def, mul_def, add_comm] | lemma | sym_alg.mul_comm | algebra | src/algebra/symmetrized.lean | [
"algebra.jordan.basic",
"algebra.module.basic"
] | [
"add_comm_semigroup",
"invertible",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
triv_sq_zero_ext (R : Type u) (M : Type v) | R × M | def | triv_sq_zero_ext | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | "Trivial Square-Zero Extension".
Given a module `M` over a ring `R`, the trivial square-zero extension of `M` over `R` is defined
to be the `R`-algebra `R × M` with multiplication given by
`(r₁ + m₁) * (r₂ + m₂) = r₁ r₂ + r₁ m₂ + r₂ m₁`.
It is a square-zero extension because `M^2 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl [has_zero M] (r : R) : tsze R M | (r, 0) | def | triv_sq_zero_ext.inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | The canonical inclusion `R → triv_sq_zero_ext R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr [has_zero R] (m : M) : tsze R M | (0, m) | def | triv_sq_zero_ext.inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | The canonical inclusion `M → triv_sq_zero_ext R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst (x : tsze R M) : R | x.1 | def | triv_sq_zero_ext.fst | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | The canonical projection `triv_sq_zero_ext R M → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd (x : tsze R M) : M | x.2 | def | triv_sq_zero_ext.snd | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | The canonical projection `triv_sq_zero_ext R M → M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_mk (r : R) (m : M) : fst (r, m) = r | rfl | lemma | triv_sq_zero_ext.fst_mk | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_mk (r : R) (m : M) : snd (r, m) = m | rfl | lemma | triv_sq_zero_ext.snd_mk | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {x y : tsze R M} (h1 : x.fst = y.fst) (h2 : x.snd = y.snd) : x = y | prod.ext h1 h2 | lemma | triv_sq_zero_ext.ext | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"prod.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_inl [has_zero M] (r : R) : (inl r : tsze R M).fst = r | rfl | lemma | triv_sq_zero_ext.fst_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_inl [has_zero M] (r : R) : (inl r : tsze R M).snd = 0 | rfl | lemma | triv_sq_zero_ext.snd_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_inl [has_zero M] : fst ∘ (inl : R → tsze R M) = id | rfl | lemma | triv_sq_zero_ext.fst_comp_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_inl [has_zero M] : snd ∘ (inl : R → tsze R M) = 0 | rfl | lemma | triv_sq_zero_ext.snd_comp_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_inr [has_zero R] (m : M) : (inr m : tsze R M).fst = 0 | rfl | lemma | triv_sq_zero_ext.fst_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_inr [has_zero R] (m : M) : (inr m : tsze R M).snd = m | rfl | lemma | triv_sq_zero_ext.snd_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_inr [has_zero R] : fst ∘ (inr : M → tsze R M) = 0 | rfl | lemma | triv_sq_zero_ext.fst_comp_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_inr [has_zero R] : snd ∘ (inr : M → tsze R M) = id | rfl | lemma | triv_sq_zero_ext.snd_comp_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_injective [has_zero M] : function.injective (inl : R → tsze R M) | function.left_inverse.injective $ fst_inl _ | lemma | triv_sq_zero_ext.inl_injective | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_injective [has_zero R] : function.injective (inr : M → tsze R M) | function.left_inverse.injective $ snd_inr _ | lemma | triv_sq_zero_ext.inr_injective | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_zero [has_zero R] [has_zero M] : (0 : tsze R M).fst = 0 | rfl | lemma | triv_sq_zero_ext.fst_zero | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_zero [has_zero R] [has_zero M] : (0 : tsze R M).snd = 0 | rfl | lemma | triv_sq_zero_ext.snd_zero | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_add [has_add R] [has_add M] (x₁ x₂ : tsze R M) :
(x₁ + x₂).fst = x₁.fst + x₂.fst | rfl | lemma | triv_sq_zero_ext.fst_add | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_add [has_add R] [has_add M] (x₁ x₂ : tsze R M) :
(x₁ + x₂).snd = x₁.snd + x₂.snd | rfl | lemma | triv_sq_zero_ext.snd_add | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_neg [has_neg R] [has_neg M] (x : tsze R M) : (-x).fst = -x.fst | rfl | lemma | triv_sq_zero_ext.fst_neg | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_neg [has_neg R] [has_neg M] (x : tsze R M) : (-x).snd = -x.snd | rfl | lemma | triv_sq_zero_ext.snd_neg | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_sub [has_sub R] [has_sub M] (x₁ x₂ : tsze R M) :
(x₁ - x₂).fst = x₁.fst - x₂.fst | rfl | lemma | triv_sq_zero_ext.fst_sub | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_sub [has_sub R] [has_sub M] (x₁ x₂ : tsze R M) :
(x₁ - x₂).snd = x₁.snd - x₂.snd | rfl | lemma | triv_sq_zero_ext.snd_sub | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_smul [has_smul S R] [has_smul S M] (s : S) (x : tsze R M) :
(s • x).fst = s • x.fst | rfl | lemma | triv_sq_zero_ext.fst_smul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_smul [has_smul S R] [has_smul S M] (s : S) (x : tsze R M) :
(s • x).snd = s • x.snd | rfl | lemma | triv_sq_zero_ext.snd_smul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_sum {ι} [add_comm_monoid R] [add_comm_monoid M] (s : finset ι) (f : ι → tsze R M) :
(∑ i in s, f i).fst = ∑ i in s, (f i).fst | prod.fst_sum | lemma | triv_sq_zero_ext.fst_sum | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_sum {ι} [add_comm_monoid R] [add_comm_monoid M] (s : finset ι) (f : ι → tsze R M) :
(∑ i in s, f i).snd = ∑ i in s, (f i).snd | prod.snd_sum | lemma | triv_sq_zero_ext.snd_sum | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_zero [has_zero R] [has_zero M] : (inl 0 : tsze R M) = 0 | rfl | lemma | triv_sq_zero_ext.inl_zero | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_add [has_add R] [add_zero_class M] (r₁ r₂ : R) :
(inl (r₁ + r₂) : tsze R M) = inl r₁ + inl r₂ | ext rfl (add_zero 0).symm | lemma | triv_sq_zero_ext.inl_add | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_neg [has_neg R] [sub_neg_zero_monoid M] (r : R) :
(inl (-r) : tsze R M) = -inl r | ext rfl neg_zero.symm | lemma | triv_sq_zero_ext.inl_neg | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"sub_neg_zero_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_sub [has_sub R] [sub_neg_zero_monoid M] (r₁ r₂ : R) :
(inl (r₁ - r₂) : tsze R M) = inl r₁ - inl r₂ | ext rfl (sub_zero _).symm | lemma | triv_sq_zero_ext.inl_sub | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"sub_neg_zero_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_smul [monoid S] [add_monoid M] [has_smul S R] [distrib_mul_action S M]
(s : S) (r : R) : (inl (s • r) : tsze R M) = s • inl r | ext rfl (smul_zero s).symm | lemma | triv_sq_zero_ext.inl_smul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"has_smul",
"monoid",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_sum {ι} [add_comm_monoid R] [add_comm_monoid M] (s : finset ι) (f : ι → R) :
(inl (∑ i in s, f i) : tsze R M) = ∑ i in s, inl (f i) | (linear_map.inl ℕ _ _).map_sum | lemma | triv_sq_zero_ext.inl_sum | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"finset",
"linear_map.inl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_zero [has_zero R] [has_zero M] : (inr 0 : tsze R M) = 0 | rfl | lemma | triv_sq_zero_ext.inr_zero | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_add [add_zero_class R] [add_zero_class M] (m₁ m₂ : M) :
(inr (m₁ + m₂) : tsze R M) = inr m₁ + inr m₂ | ext (add_zero 0).symm rfl | lemma | triv_sq_zero_ext.inr_add | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_neg [sub_neg_zero_monoid R] [has_neg M] (m : M) :
(inr (-m) : tsze R M) = -inr m | ext neg_zero.symm rfl | lemma | triv_sq_zero_ext.inr_neg | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"sub_neg_zero_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_sub [sub_neg_zero_monoid R] [has_sub M] (m₁ m₂ : M) :
(inr (m₁ - m₂) : tsze R M) = inr m₁ - inr m₂ | ext (sub_zero _).symm rfl | lemma | triv_sq_zero_ext.inr_sub | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"sub_neg_zero_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_smul [has_zero R] [has_zero S] [smul_with_zero S R] [has_smul S M]
(r : S) (m : M) : (inr (r • m) : tsze R M) = r • inr m | ext (smul_zero _).symm rfl | lemma | triv_sq_zero_ext.inr_smul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"has_smul",
"smul_with_zero",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_sum {ι} [add_comm_monoid R] [add_comm_monoid M] (s : finset ι) (f : ι → M) :
(inr (∑ i in s, f i) : tsze R M) = ∑ i in s, inr (f i) | (linear_map.inr ℕ _ _).map_sum | lemma | triv_sq_zero_ext.inr_sum | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"finset",
"linear_map.inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_fst_add_inr_snd_eq [add_zero_class R] [add_zero_class M] (x : tsze R M) :
inl x.fst + inr x.snd = x | ext (add_zero x.1) (zero_add x.2) | lemma | triv_sq_zero_ext.inl_fst_add_inr_snd_eq | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ind {R M} [add_zero_class R] [add_zero_class M] {P : triv_sq_zero_ext R M → Prop}
(h : ∀ r m, P (inl r + inr m)) (x) : P x | inl_fst_add_inr_snd_eq x ▸ h x.1 x.2 | lemma | triv_sq_zero_ext.ind | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_zero_class",
"triv_sq_zero_ext"
] | To show a property hold on all `triv_sq_zero_ext R M` it suffices to show it holds
on terms of the form `inl r + inr m`.
This can be used as `induction x using triv_sq_zero_ext.ind`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_ext {N} [semiring S] [add_comm_monoid R] [add_comm_monoid M] [add_comm_monoid N]
[module S R] [module S M] [module S N] ⦃f g : tsze R M →ₗ[S] N⦄
(hl : ∀ r, f (inl r) = g (inl r)) (hr : ∀ m, f (inr m) = g (inr m)) :
f = g | linear_map.prod_ext (linear_map.ext hl) (linear_map.ext hr) | lemma | triv_sq_zero_ext.linear_map_ext | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"linear_map.ext",
"linear_map.prod_ext",
"module",
"semiring"
] | This cannot be marked `@[ext]` as it ends up being used instead of `linear_map.prod_ext` when
working with `R × M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr_hom [semiring R] [add_comm_monoid M] [module R M] : M →ₗ[R] tsze R M | { to_fun := inr, ..linear_map.inr R R M } | def | triv_sq_zero_ext.inr_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"linear_map.inr",
"module",
"semiring"
] | The canonical `R`-linear inclusion `M → triv_sq_zero_ext R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_hom [semiring R] [add_comm_monoid M] [module R M] : tsze R M →ₗ[R] M | { to_fun := snd, ..linear_map.snd _ _ _ } | def | triv_sq_zero_ext.snd_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"linear_map.snd",
"module",
"semiring"
] | The canonical `R`-linear projection `triv_sq_zero_ext R M → M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_one [has_one R] [has_zero M] : (1 : tsze R M).fst = 1 | rfl | lemma | triv_sq_zero_ext.fst_one | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_one [has_one R] [has_zero M] : (1 : tsze R M).snd = 0 | rfl | lemma | triv_sq_zero_ext.snd_one | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_mul [has_mul R] [has_add M] [has_smul R M] [has_smul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) :
(x₁ * x₂).fst = x₁.fst * x₂.fst | rfl | lemma | triv_sq_zero_ext.fst_mul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_mul [has_mul R] [has_add M] [has_smul R M] [has_smul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) :
(x₁ * x₂).snd = x₁.fst • x₂.snd + op x₂.fst • x₁.snd | rfl | lemma | triv_sq_zero_ext.snd_mul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_one [has_one R] [has_zero M] : (inl 1 : tsze R M) = 1 | rfl | lemma | triv_sq_zero_ext.inl_one | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_mul [monoid R] [add_monoid M]
[distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M] (r₁ r₂ : R) :
(inl (r₁ * r₂) : tsze R M) = inl r₁ * inl r₂ | ext rfl $ show (0 : M) = r₁ • 0 + op r₂ • 0, by rw [smul_zero, zero_add, smul_zero] | lemma | triv_sq_zero_ext.inl_mul | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"monoid",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_mul_inl [monoid R] [add_monoid M]
[distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M] (r₁ r₂ : R) :
(inl r₁ * inl r₂ : tsze R M) = inl (r₁ * r₂) | (inl_mul M r₁ r₂).symm | lemma | triv_sq_zero_ext.inl_mul_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_mul_inr [semiring R] [add_comm_monoid M] [module R M] [module Rᵐᵒᵖ M]
(m₁ m₂ : M) :
(inr m₁ * inr m₂ : tsze R M) = 0 | ext (mul_zero _) $ show (0 : R) • m₂ + (0 : Rᵐᵒᵖ) • m₁ = 0, by rw [zero_smul, zero_add, zero_smul] | lemma | triv_sq_zero_ext.inr_mul_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"module",
"mul_zero",
"semiring",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_mul_inr [semiring R] [add_comm_monoid M] [module R M] [module Rᵐᵒᵖ M] (r : R) (m : M) :
(inl r * inr m : tsze R M) = inr (r • m) | ext (mul_zero r) $ show r • m + (0 : Rᵐᵒᵖ) • 0 = r • m, by rw [smul_zero, add_zero] | lemma | triv_sq_zero_ext.inl_mul_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"module",
"mul_zero",
"semiring",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_mul_inl [semiring R] [add_comm_monoid M] [module R M] [module Rᵐᵒᵖ M] (r : R) (m : M) :
(inr m * inl r : tsze R M) = inr (op r • m) | ext (zero_mul r) $ show (0 : R) • 0 + op r • m = op r • m, by rw [smul_zero, zero_add] | lemma | triv_sq_zero_ext.inr_mul_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"module",
"semiring",
"smul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_nat_cast [add_monoid_with_one R] [add_monoid M] (n : ℕ) :
(n : tsze R M).fst = n | rfl | lemma | triv_sq_zero_ext.fst_nat_cast | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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