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 |
|---|---|---|---|---|---|---|---|---|---|---|
pow_induction_on_left
{C : A → Prop}
(hr : ∀ r : R, C (algebra_map _ _ r))
(hadd : ∀ x y, C x → C y → C (x + y))
(hmul : ∀ (m ∈ M) x, C x → C (m * x))
{x : A} {n : ℕ} (hx : x ∈ M ^ n) : C x | submodule.pow_induction_on_left' M
(by exact hr) (λ x y i hx hy, hadd x y) (λ m hm i x hx, hmul _ hm _) hx | theorem | submodule.pow_induction_on_left | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"submodule.pow_induction_on_left'"
] | To show a property on elements of `M ^ n` holds, it suffices to show that it holds for scalars,
is closed under addition, and holds for `m * x` where `m ∈ M` and it holds for `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_induction_on_right
{C : A → Prop}
(hr : ∀ r : R, C (algebra_map _ _ r))
(hadd : ∀ x y, C x → C y → C (x + y))
(hmul : ∀ x, C x → ∀ (m ∈ M), C (x * m))
{x : A} {n : ℕ} (hx : x ∈ M ^ n) : C x | submodule.pow_induction_on_right' M
(by exact hr) (λ x y i hx hy, hadd x y) (λ i x hx, hmul _) hx | theorem | submodule.pow_induction_on_right | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra_map",
"submodule.pow_induction_on_right'"
] | To show a property on elements of `M ^ n` holds, it suffices to show that it holds for scalars,
is closed under addition, and holds for `x * m` where `m ∈ M` and it holds for `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_hom {A'} [semiring A'] [algebra R A'] (f : A →ₐ[R] A') :
submodule R A →*₀ submodule R A' | { to_fun := map f.to_linear_map,
map_zero' := submodule.map_bot _,
map_one' := submodule.map_one _,
map_mul' := λ _ _, submodule.map_mul _ _ _} | def | submodule.map_hom | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra",
"semiring",
"submodule",
"submodule.map_bot",
"submodule.map_mul",
"submodule.map_one"
] | `submonoid.map` as a `monoid_with_zero_hom`, when applied to `alg_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_opposite : submodule R Aᵐᵒᵖ ≃+* (submodule R A)ᵐᵒᵖ | { to_fun := λ p, op $ p.comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ),
inv_fun := λ p, p.unop.comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A),
left_inv := λ p, set_like.coe_injective $ rfl,
right_inv := λ p, unop_injective $ set_like.coe_injective rfl,
map_add' := λ p q, by simp [co... | def | submodule.equiv_opposite | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"inv_fun",
"set_like.coe_injective",
"submodule"
] | The ring of submodules of the opposite algebra is isomorphic to the opposite ring of
submodules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pow {A'} [semiring A'] [algebra R A'] (f : A →ₐ[R] A') (n : ℕ) :
map f.to_linear_map (M ^ n) = map f.to_linear_map M ^ n | map_pow (map_hom f) M n | lemma | submodule.map_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra",
"map_pow",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_unop_pow (n : ℕ) :
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (M ^ n) =
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) M ^ n | (equiv_opposite : submodule R Aᵐᵒᵖ ≃+* _).symm.map_pow (op M) n | lemma | submodule.comap_unop_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_op_pow (n : ℕ) (M : submodule R Aᵐᵒᵖ) :
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (M ^ n) =
comap (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) M ^ n | op_injective $ (equiv_opposite : submodule R Aᵐᵒᵖ ≃+* _).map_pow M n | lemma | submodule.comap_op_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"map_pow",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_op_pow (n : ℕ) :
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (M ^ n) =
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) M ^ n | by rw [map_equiv_eq_comap_symm, map_equiv_eq_comap_symm, comap_unop_pow] | lemma | submodule.map_op_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unop_pow (n : ℕ) (M : submodule R Aᵐᵒᵖ) :
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (M ^ n) =
map (↑(op_linear_equiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) M ^ n | by rw [←comap_equiv_eq_map_symm, ←comap_equiv_eq_map_symm, comap_op_pow] | lemma | submodule.map_unop_pow | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span.ring_hom : set_semiring A →+* submodule R A | { to_fun := λ s, submodule.span R s.down,
map_zero' := span_empty,
map_one' := one_eq_span.symm,
map_add' := span_union,
map_mul' := λ s t, by rw [set_semiring.down_mul, span_mul_span, ← image_mul_prod] } | def | submodule.span.ring_hom | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"set_semiring",
"set_semiring.down_mul",
"submodule",
"submodule.span"
] | `span` is a semiring homomorphism (recall multiplication is pointwise multiplication of subsets
on either side). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pointwise_mul_semiring_action : mul_semiring_action α (submodule R A) | { smul_mul := λ r x y, submodule.map_mul x y $ mul_semiring_action.to_alg_hom R A r,
smul_one := λ r, submodule.map_one $ mul_semiring_action.to_alg_hom R A r,
..submodule.pointwise_distrib_mul_action } | def | submodule.pointwise_mul_semiring_action | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_semiring_action",
"mul_semiring_action.to_alg_hom",
"submodule",
"submodule.map_mul",
"submodule.map_one",
"submodule.pointwise_distrib_mul_action"
] | The action on a submodule corresponding to applying the action to every element.
This is available as an instance in the `pointwise` locale.
This is a stronger version of `submodule.pointwise_distrib_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mem_mul_rev (hm : m ∈ M) (hn : n ∈ N) : n * m ∈ M * N | mul_comm m n ▸ mul_mem_mul hm hn | theorem | submodule.mul_mem_mul_rev | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm : M * N = N * M | le_antisymm (mul_le.2 $ λ r hrm s hsn, mul_mem_mul_rev hsn hrm)
(mul_le.2 $ λ r hrn s hsm, mul_mem_mul_rev hsm hrn) | theorem | submodule.mul_comm | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_span {ι : Type*} (s : finset ι) (M : ι → set A) :
(∏ i in s, submodule.span R (M i)) = submodule.span R (∏ i in s, M i) | begin
letI := classical.dec_eq ι,
refine finset.induction_on s _ _,
{ simp [one_eq_span, set.singleton_one] },
{ intros _ _ H ih,
rw [finset.prod_insert H, finset.prod_insert H, ih, span_mul_span] }
end | lemma | submodule.prod_span | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"classical.dec_eq",
"finset",
"finset.induction_on",
"finset.prod_insert",
"ih",
"set.singleton_one",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_span_singleton {ι : Type*} (s : finset ι) (x : ι → A) :
(∏ i in s, span R ({x i} : set A)) = span R {∏ i in s, x i} | by rw [prod_span, set.finset_prod_singleton] | lemma | submodule.prod_span_singleton | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"finset",
"set.finset_prod_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_set : module (set_semiring A) (submodule R A) | { smul := λ s P, span R s.down * P,
smul_add := λ _ _ _, mul_add _ _ _,
add_smul := λ s t P,
by simp_rw [has_smul.smul, set_semiring.down_add, span_union, sup_mul, add_eq_sup],
mul_smul := λ s t P,
by simp_rw [has_smul.smul, set_semiring.down_mul, ← mul_assoc, span_mul_span],
one_smul := λ P,
by sim... | instance | submodule.module_set | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"add_eq_sup",
"add_smul",
"module",
"mul_assoc",
"one_mul",
"one_smul",
"set_semiring",
"set_semiring.down_add",
"set_semiring.down_mul",
"set_semiring.down_one",
"set_semiring.down_zero",
"smul_add",
"smul_zero",
"submodule",
"sup_mul",
"zero_smul"
] | R-submodules of the R-algebra A are a module over `set A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (s : set_semiring A) (P : submodule R A) : s • P = span R s.down * P | rfl | lemma | submodule.smul_def | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"set_semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul {s t : set_semiring A} {M N : submodule R A} (h₁ : s.down ⊆ t.down)
(h₂ : M ≤ N) : s • M ≤ t • N | mul_le_mul (span_mono h₁) h₂ | lemma | submodule.smul_le_smul | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_le_mul",
"set_semiring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_singleton (a : A) (M : submodule R A) :
({a} : set A).up • M = M.map (linear_map.mul_left R a) | begin
conv_lhs {rw ← span_eq M},
change span _ _ * span _ _ = _,
rw [span_mul_span],
apply le_antisymm,
{ rw span_le,
rintros _ ⟨b, m, hb, hm, rfl⟩,
rw [set_like.mem_coe, mem_map, set.mem_singleton_iff.mp hb],
exact ⟨m, hm, rfl⟩ },
{ rintros _ ⟨m, hm, rfl⟩, exact subset_span ⟨a, m, set.mem_singl... | lemma | submodule.smul_singleton | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"linear_map.mul_left",
"mem_map",
"set.mem_singleton",
"set_like.mem_coe",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_div_iff_forall_mul_mem {x : A} {I J : submodule R A} :
x ∈ I / J ↔ ∀ y ∈ J, x * y ∈ I | iff.refl _ | lemma | submodule.mem_div_iff_forall_mul_mem | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_div_iff_smul_subset {x : A} {I J : submodule R A} : x ∈ I / J ↔ x • (J : set A) ⊆ I | ⟨ λ h y ⟨y', hy', xy'_eq_y⟩, by { rw ← xy'_eq_y, apply h, assumption },
λ h y hy, h (set.smul_mem_smul_set hy) ⟩ | lemma | submodule.mem_div_iff_smul_subset | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"set.smul_mem_smul_set",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff {I J K : submodule R A} : I ≤ J / K ↔ ∀ (x ∈ I) (z ∈ K), x * z ∈ J | iff.refl _ | lemma | submodule.le_div_iff | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"le_div_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_mul_le {I J K : submodule R A} : I ≤ J / K ↔ I * K ≤ J | by rw [le_div_iff, mul_le] | lemma | submodule.le_div_iff_mul_le | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"le_div_iff",
"le_div_iff_mul_le",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_one_div {I : submodule R A} :
1 ≤ 1 / I ↔ I ≤ 1 | begin
split, all_goals {intro hI},
{rwa [le_div_iff_mul_le, one_mul] at hI},
{rwa [le_div_iff_mul_le, one_mul]},
end | lemma | submodule.one_le_one_div | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"le_div_iff_mul_le",
"one_le_one_div",
"one_mul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_mul_one_div {I : submodule R A} (hI : I ≤ 1) :
I ≤ I * (1 / I) | begin
rw [← mul_one I] {occs := occurrences.pos [1]},
apply mul_le_mul_right (one_le_one_div.mpr hI),
end | lemma | submodule.le_self_mul_one_div | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_le_mul_right",
"mul_one",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_div_le_one {I : submodule R A} : I * (1 / I) ≤ 1 | begin
rw submodule.mul_le,
intros m hm n hn,
rw [submodule.mem_div_iff_forall_mul_mem] at hn,
rw mul_comm,
exact hn m hm,
end | lemma | submodule.mul_one_div_le_one | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"mul_comm",
"submodule",
"submodule.mem_div_iff_forall_mul_mem",
"submodule.mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div {B : Type*} [comm_semiring B] [algebra R B]
(I J : submodule R A) (h : A ≃ₐ[R] B) :
(I / J).map h.to_linear_map = I.map h.to_linear_map / J.map h.to_linear_map | begin
ext x,
simp only [mem_map, mem_div_iff_forall_mul_mem],
split,
{ rintro ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩,
exact ⟨x * y, hx _ hy, h.map_mul x y⟩ },
{ rintro hx,
refine ⟨h.symm x, λ z hz, _, h.apply_symm_apply x⟩,
obtain ⟨xz, xz_mem, hxz⟩ := hx (h z) ⟨z, hz, rfl⟩,
convert xz_mem,
apply h.in... | lemma | submodule.map_div | algebra.algebra | src/algebra/algebra/operations.lean | [
"algebra.algebra.bilinear",
"algebra.algebra.equiv",
"algebra.module.submodule.pointwise",
"algebra.module.submodule.bilinear",
"algebra.module.opposites",
"algebra.order.kleene",
"data.finset.pointwise",
"data.set.semiring",
"data.set.pointwise.big_operators",
"group_theory.group_action.sub_mul_a... | [
"algebra",
"comm_semiring",
"map_div",
"mem_map",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra {r : comm_semiring R}
[s : ∀ i, semiring (f i)] [∀ i, algebra R (f i)] :
algebra R (Π i : I, f i) | { commutes' := λ a f, begin ext, simp [algebra.commutes], end,
smul_def' := λ a f, begin ext, simp [algebra.smul_def], end,
..(pi.ring_hom (λ i, algebra_map R (f i)) : R →+* Π i : I, f i) } | instance | pi.algebra | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"algebra.commutes",
"algebra.smul_def",
"algebra_map",
"comm_semiring",
"pi.ring_hom",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_def {r : comm_semiring R}
[s : ∀ i, semiring (f i)] [∀ i, algebra R (f i)] (a : R) :
algebra_map R (Π i, f i) a = (λ i, algebra_map R (f i) a) | rfl | lemma | pi.algebra_map_def | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply {r : comm_semiring R}
[s : ∀ i, semiring (f i)] [∀ i, algebra R (f i)] (a : R) (i : I) :
algebra_map R (Π i, f i) a i = algebra_map R (f i) a | rfl | lemma | pi.algebra_map_apply | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"algebra_map",
"algebra_map_apply",
"comm_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_alg_hom {r : comm_semiring R} [Π i, semiring (f i)] [Π i, algebra R (f i)] (i : I) :
(Π i, f i) →ₐ[R] f i | { to_fun := λ f, f i, commutes' := λ r, rfl, .. pi.eval_ring_hom f i} | def | pi.eval_alg_hom | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"comm_semiring",
"pi.eval_ring_hom",
"semiring"
] | `function.eval` as an `alg_hom`. The name matches `pi.eval_ring_hom`, `pi.eval_monoid_hom`,
etc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_alg_hom : B →ₐ[R] (A → B) | { to_fun := function.const _,
commutes' := λ r, rfl,
.. pi.const_ring_hom A B} | def | pi.const_alg_hom | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"pi.const_ring_hom"
] | `function.const` as an `alg_hom`. The name matches `pi.const_ring_hom`, `pi.const_monoid_hom`,
etc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_ring_hom_eq_algebra_map : const_ring_hom A R = algebra_map R (A → R) | rfl | lemma | pi.const_ring_hom_eq_algebra_map | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra_map"
] | When `R` is commutative and permits an `algebra_map`, `pi.const_ring_hom` is equal to that
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_alg_hom_eq_algebra_of_id : const_alg_hom R A R = algebra.of_id R (A → R) | rfl | lemma | pi.const_alg_hom_eq_algebra_of_id | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra.of_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.algebra {R : Type*} (I : Type*) (A : Type*) [comm_semiring R]
[semiring A] [algebra R A] : algebra R (I → A) | pi.algebra _ _ | instance | function.algebra | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"comm_semiring",
"pi.algebra",
"semiring"
] | A special case of `pi.algebra` for non-dependent types. Lean struggles to elaborate
definitions elsewhere in the library without this, | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_left (f : A →ₐ[R] B) (I : Type*) : (I → A) →ₐ[R] (I → B) | { to_fun := λ h, f ∘ h,
commutes' := λ c, by { ext, exact f.commutes' c },
.. f.to_ring_hom.comp_left I } | def | alg_hom.comp_left | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [] | `R`-algebra homomorphism between the function spaces `I → A` and `I → B`, induced by an
`R`-algebra homomorphism `f` between `A` and `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right {R ι : Type*} {A₁ A₂ : ι → Type*} [comm_semiring R]
[Π i, semiring (A₁ i)] [Π i, semiring (A₂ i)] [Π i, algebra R (A₁ i)] [Π i, algebra R (A₂ i)]
(e : Π i, A₁ i ≃ₐ[R] A₂ i) : (Π i, A₁ i) ≃ₐ[R] Π i, A₂ i | { to_fun := λ x j, e j (x j),
inv_fun := λ x j, (e j).symm (x j),
commutes' := λ r, by { ext i, simp },
.. @ring_equiv.Pi_congr_right ι A₁ A₂ _ _ (λ i, (e i).to_ring_equiv) } | def | alg_equiv.Pi_congr_right | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"comm_semiring",
"inv_fun",
"ring_equiv.Pi_congr_right",
"semiring"
] | A family of algebra equivalences `Π j, (A₁ j ≃ₐ A₂ j)` generates a
multiplicative equivalence between `Π j, A₁ j` and `Π j, A₂ j`.
This is the `alg_equiv` version of `equiv.Pi_congr_right`, and the dependent version of
`alg_equiv.arrow_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right_refl {R ι : Type*} {A : ι → Type*} [comm_semiring R]
[Π i, semiring (A i)] [Π i, algebra R (A i)] :
Pi_congr_right (λ i, (alg_equiv.refl : A i ≃ₐ[R] A i)) = alg_equiv.refl | rfl | lemma | alg_equiv.Pi_congr_right_refl | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"alg_equiv.refl",
"algebra",
"comm_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right_symm {R ι : Type*} {A₁ A₂ : ι → Type*} [comm_semiring R]
[Π i, semiring (A₁ i)] [Π i, semiring (A₂ i)] [Π i, algebra R (A₁ i)] [Π i, algebra R (A₂ i)]
(e : Π i, A₁ i ≃ₐ[R] A₂ i) : (Pi_congr_right e).symm = (Pi_congr_right $ λ i, (e i).symm) | rfl | lemma | alg_equiv.Pi_congr_right_symm | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"comm_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right_trans {R ι : Type*} {A₁ A₂ A₃ : ι → Type*} [comm_semiring R]
[Π i, semiring (A₁ i)] [Π i, semiring (A₂ i)] [Π i, semiring (A₃ i)]
[Π i, algebra R (A₁ i)] [Π i, algebra R (A₂ i)] [Π i, algebra R (A₃ i)]
(e₁ : Π i, A₁ i ≃ₐ[R] A₂ i) (e₂ : Π i, A₂ i ≃ₐ[R] A₃ i) :
(Pi_congr_right e₁).trans (Pi_congr_r... | rfl | lemma | alg_equiv.Pi_congr_right_trans | algebra.algebra | src/algebra/algebra/pi.lean | [
"algebra.algebra.equiv"
] | [
"algebra",
"comm_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra : algebra R (A × B) | { commutes' := by { rintro r ⟨a, b⟩, dsimp, rw [commutes r a, commutes r b] },
smul_def' := by { rintro r ⟨a, b⟩, dsimp, rw [algebra.smul_def r a, algebra.smul_def r b] },
.. prod.module,
.. ring_hom.prod (algebra_map R A) (algebra_map R B) } | instance | prod.algebra | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"ring_hom.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply (r : R) :
algebra_map R (A × B) r = (algebra_map R A r, algebra_map R B r) | rfl | lemma | prod.algebra_map_apply | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"algebra_map",
"algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : A × B →ₐ[R] A | { commutes' := λ r, rfl, .. ring_hom.fst A B} | def | alg_hom.fst | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"ring_hom.fst"
] | First projection as `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : A × B →ₐ[R] B | { commutes' := λ r, rfl, .. ring_hom.snd A B} | def | alg_hom.snd | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"ring_hom.snd"
] | Second projection as `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : (A →ₐ[R] B × C) | { commutes' := λ r, by simp only [to_ring_hom_eq_coe, ring_hom.to_fun_eq_coe, ring_hom.prod_apply,
coe_to_ring_hom, commutes, prod.algebra_map_apply],
.. (f.to_ring_hom.prod g.to_ring_hom) } | def | alg_hom.prod | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"prod.algebra_map_apply",
"ring_hom.prod_apply",
"ring_hom.to_fun_eq_coe"
] | The `pi.prod` of two morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) : ⇑(f.prod g) = pi.prod f g | rfl | lemma | alg_hom.coe_prod | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"pi.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) :
(fst R B C).comp (prod f g) = f | by ext; refl | theorem | alg_hom.fst_prod | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_prod (f : A →ₐ[R] B) (g : A →ₐ[R] C) :
(snd R B C).comp (prod f g) = g | by ext; refl | theorem | alg_hom.snd_prod | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_fst_snd : prod (fst R A B) (snd R A B) = 1 | fun_like.coe_injective pi.prod_fst_snd | theorem | alg_hom.prod_fst_snd | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"fun_like.coe_injective",
"pi.prod_fst_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv : ((A →ₐ[R] B) × (A →ₐ[R] C)) ≃ (A →ₐ[R] B × C) | { to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((fst _ _ _).comp f, (snd _ _ _).comp f),
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl } | def | alg_hom.prod_equiv | algebra.algebra | src/algebra/algebra/prod.lean | [
"algebra.algebra.hom"
] | [
"inv_fun"
] | Taking the product of two maps with the same domain is equivalent to taking the product of
their codomains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars (R S M : Type*) : Type* | M | def | restrict_scalars | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [] | If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the
category of `S`-modules to the category of representations of the algebra `S` (over `R`). The type
synonym `restrict_scalars` is essentially this equivalence.
Warning: use this type synonym judiciously! Consider an example wher... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.module_orig [I : module S M] :
module S (restrict_scalars R S M) | I | def | restrict_scalars.module_orig | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"module",
"restrict_scalars"
] | We temporarily install an action of the original ring on `restrict_sclars R S M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.op_module [module Sᵐᵒᵖ M] : module Rᵐᵒᵖ (restrict_scalars R S M) | begin
letI : module Sᵐᵒᵖ (restrict_scalars R S M) := ‹module Sᵐᵒᵖ M›,
exact module.comp_hom M (algebra_map R S).op
end | instance | restrict_scalars.op_module | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"module",
"module.comp_hom",
"restrict_scalars"
] | When `M` is a right-module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a
right-module structure over `R`.
The preferred way of setting this up is
`[module Rᵐᵒᵖ M] [module Sᵐᵒᵖ M] [is_scalar_tower Rᵐᵒᵖ Sᵐᵒᵖ M]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.is_central_scalar [module S M] [module Sᵐᵒᵖ M] [is_central_scalar S M] :
is_central_scalar R (restrict_scalars R S M) | { op_smul_eq_smul := λ r x, (op_smul_eq_smul (algebra_map R S r) (_ : M) : _)} | instance | restrict_scalars.is_central_scalar | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"is_central_scalar",
"module",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.lsmul [module S M] : S →ₐ[R] module.End R (restrict_scalars R S M) | begin
-- We use `restrict_scalars.module_orig` in the implementation,
-- but not in the type.
letI : module S (restrict_scalars R S M) := restrict_scalars.module_orig R S M,
exact algebra.lsmul R (restrict_scalars R S M),
end | def | restrict_scalars.lsmul | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra.lsmul",
"module",
"module.End",
"restrict_scalars",
"restrict_scalars.module_orig"
] | The `R`-algebra homomorphism from the original coefficient algebra `S` to endomorphisms
of `restrict_scalars R S M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.add_equiv : restrict_scalars R S M ≃+ M | add_equiv.refl M | def | restrict_scalars.add_equiv | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"restrict_scalars"
] | `restrict_scalars.add_equiv` is the additive equivalence with the original module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.add_equiv_map_smul (c : R) (x : restrict_scalars R S M) :
restrict_scalars.add_equiv R S M (c • x)
= (algebra_map R S c) • restrict_scalars.add_equiv R S M x | rfl | lemma | restrict_scalars.add_equiv_map_smul | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"restrict_scalars",
"restrict_scalars.add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.smul_def (c : R) (x : restrict_scalars R S M) :
c • x = (restrict_scalars.add_equiv R S M).symm
(algebra_map R S c • restrict_scalars.add_equiv R S M x) | rfl | lemma | restrict_scalars.smul_def | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"restrict_scalars",
"restrict_scalars.add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.add_equiv_symm_map_algebra_map_smul (r : R) (x : M) :
(restrict_scalars.add_equiv R S M).symm (algebra_map R S r • x)
= r • (restrict_scalars.add_equiv R S M).symm x | rfl | lemma | restrict_scalars.add_equiv_symm_map_algebra_map_smul | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"restrict_scalars.add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.add_equiv_symm_map_smul_smul (r : R) (s : S) (x : M) :
(restrict_scalars.add_equiv R S M).symm ((r • s) • x)
= r • (restrict_scalars.add_equiv R S M ).symm (s • x) | by { rw [algebra.smul_def, mul_smul], refl, } | lemma | restrict_scalars.add_equiv_symm_map_smul_smul | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra.smul_def",
"restrict_scalars.add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.lsmul_apply_apply (s : S) (x : restrict_scalars R S M) :
restrict_scalars.lsmul R S M s x =
(restrict_scalars.add_equiv R S M).symm (s • (restrict_scalars.add_equiv R S M x)) | rfl | lemma | restrict_scalars.lsmul_apply_apply | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"restrict_scalars",
"restrict_scalars.add_equiv",
"restrict_scalars.lsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.ring_equiv : restrict_scalars R S A ≃+* A | ring_equiv.refl _ | def | restrict_scalars.ring_equiv | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"restrict_scalars",
"ring_equiv.refl"
] | Tautological ring isomorphism `restrict_scalars R S A ≃+* A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars.ring_equiv_map_smul (r : R) (x : restrict_scalars R S A) :
restrict_scalars.ring_equiv R S A (r • x)
= (algebra_map R S r) • restrict_scalars.ring_equiv R S A x | rfl | lemma | restrict_scalars.ring_equiv_map_smul | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"restrict_scalars",
"restrict_scalars.ring_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars.ring_equiv_algebra_map (r : R) :
restrict_scalars.ring_equiv R S A (algebra_map R (restrict_scalars R S A) r) =
algebra_map S A (algebra_map R S r) | rfl | lemma | restrict_scalars.ring_equiv_algebra_map | algebra.algebra | src/algebra/algebra/restrict_scalars.lean | [
"algebra.algebra.tower"
] | [
"algebra_map",
"restrict_scalars",
"restrict_scalars.ring_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resolvent_set (a : A) : set R | { r : R | is_unit (↑ₐr - a) } | def | resolvent_set | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit"
] | Given a commutative ring `R` and an `R`-algebra `A`, the *resolvent set* of `a : A`
is the `set R` consisting of those `r : R` for which `r•1 - a` is a unit of the
algebra `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
spectrum (a : A) : set R | (resolvent_set R a)ᶜ | def | spectrum | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"resolvent_set"
] | Given a commutative ring `R` and an `R`-algebra `A`, the *spectrum* of `a : A`
is the `set R` consisting of those `r : R` for which `r•1 - a` is not a unit of the
algebra `A`.
The spectrum is simply the complement of the resolvent set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
resolvent (a : A) (r : R) : A | ring.inverse (↑ₐr - a) | def | resolvent | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"ring.inverse"
] | Given an `a : A` where `A` is an `R`-algebra, the *resolvent* is
a map `R → A` which sends `r : R` to `(algebra_map R A r - a)⁻¹` when
`r ∈ resolvent R A` and `0` when `r ∈ spectrum R A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit.sub_inv_smul {r : Rˣ} {s : R} {a : A}
(h : is_unit $ r • ↑ₐs - a) : Aˣ | { val := ↑ₐs - r⁻¹ • a,
inv := r • ↑h.unit⁻¹,
val_inv := by rw [mul_smul_comm, ←smul_mul_assoc, smul_sub, smul_inv_smul, h.mul_coe_inv],
inv_val := by rw [smul_mul_assoc, ←mul_smul_comm, smul_sub, smul_inv_smul, h.coe_inv_mul], } | def | is_unit.sub_inv_smul | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit",
"mul_smul_comm",
"smul_inv_smul",
"smul_mul_assoc",
"smul_sub"
] | The unit `1 - r⁻¹ • a` constructed from `r • 1 - a` when the latter is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_iff {r : R} {a : A} :
r ∈ σ a ↔ ¬ is_unit (↑ₐr - a) | iff.rfl | lemma | spectrum.mem_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_iff {r : R} {a : A} :
r ∉ σ a ↔ is_unit (↑ₐr - a) | by { apply not_iff_not.mp, simp [set.not_not_mem, mem_iff] } | lemma | spectrum.not_mem_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit",
"set.not_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem_iff {a : A} : (0 : R) ∈ σ a ↔ ¬is_unit a | by rw [mem_iff, map_zero, zero_sub, is_unit.neg_iff] | lemma | spectrum.zero_mem_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit",
"is_unit.neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_not_mem_iff {a : A} : (0 : R) ∉ σ a ↔ is_unit a | by rw [zero_mem_iff, not_not] | lemma | spectrum.zero_not_mem_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_resolvent_set_of_left_right_inverse {r : R} {a b c : A}
(h₁ : (↑ₐr - a) * b = 1) (h₂ : c * (↑ₐr - a) = 1) :
r ∈ resolvent_set R a | units.is_unit ⟨↑ₐr - a, b, h₁, by rwa ←left_inv_eq_right_inv h₂ h₁⟩ | lemma | spectrum.mem_resolvent_set_of_left_right_inverse | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"resolvent_set",
"units.is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_resolvent_set_iff {r : R} {a : A} :
r ∈ resolvent_set R a ↔ is_unit (↑ₐr - a) | iff.rfl | lemma | spectrum.mem_resolvent_set_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit",
"resolvent_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resolvent_set_of_subsingleton [subsingleton A] (a : A) :
resolvent_set R a = set.univ | by simp_rw [resolvent_set, subsingleton.elim (algebra_map R A _ - a) 1, is_unit_one,
set.set_of_true] | lemma | spectrum.resolvent_set_of_subsingleton | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra_map",
"is_unit_one",
"resolvent_set",
"set.set_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subsingleton [subsingleton A] (a : A) :
spectrum R a = ∅ | by rw [spectrum, resolvent_set_of_subsingleton, set.compl_univ] | lemma | spectrum.of_subsingleton | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"set.compl_univ",
"spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
resolvent_eq {a : A} {r : R} (h : r ∈ resolvent_set R a) :
resolvent a r = ↑h.unit⁻¹ | ring.inverse_unit h.unit | lemma | spectrum.resolvent_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"resolvent",
"resolvent_set",
"ring.inverse_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_smul_resolvent {r : Rˣ} {s : R} {a : A} :
r • resolvent a (s : R) = resolvent (r⁻¹ • a) (r⁻¹ • s : R) | begin
by_cases h : s ∈ spectrum R a,
{ rw [mem_iff] at h,
simp only [resolvent, algebra.algebra_map_eq_smul_one] at *,
rw [smul_assoc, ←smul_sub],
have h' : ¬ is_unit (r⁻¹ • (s • 1 - a)),
from λ hu, h (by simpa only [smul_inv_smul] using is_unit.smul r hu),
simp only [ring.inverse_non_unit _ h... | lemma | spectrum.units_smul_resolvent | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"is_unit",
"is_unit.smul",
"resolvent",
"ring.inverse_non_unit",
"ring.inverse_unit",
"smul_assoc",
"smul_inv_smul",
"smul_zero",
"spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_smul_resolvent_self {r : Rˣ} {a : A} :
r • resolvent a (r : R) = resolvent (r⁻¹ • a) (1 : R) | by simpa only [units.smul_def, algebra.id.smul_eq_mul, units.inv_mul]
using @units_smul_resolvent _ _ _ _ _ r r a | lemma | spectrum.units_smul_resolvent_self | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.id.smul_eq_mul",
"resolvent",
"units.inv_mul",
"units.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_resolvent {r : R} {a : A} :
r ∈ resolvent_set R a ↔ is_unit (resolvent a r) | is_unit_ring_inverse.symm | lemma | spectrum.is_unit_resolvent | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit",
"resolvent",
"resolvent_set"
] | The resolvent is a unit when the argument is in the resolvent set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_mem_resolvent_set {r : Rˣ} {a : Aˣ} (h : (r : R) ∈ resolvent_set R (a : A)) :
(↑r⁻¹ : R) ∈ resolvent_set R (↑a⁻¹ : A) | begin
rw [mem_resolvent_set_iff, algebra.algebra_map_eq_smul_one, ←units.smul_def] at h ⊢,
rw [is_unit.smul_sub_iff_sub_inv_smul, inv_inv, is_unit.sub_iff],
have h₁ : (a : A) * (r • (↑a⁻¹ : A) - 1) = r • 1 - a,
{ rw [mul_sub, mul_smul_comm, a.mul_inv, mul_one], },
have h₂ : (r • (↑a⁻¹ : A) - 1) * a = r • 1 - ... | lemma | spectrum.inv_mem_resolvent_set | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"commute",
"inv_inv",
"is_unit.smul_sub_iff_sub_inv_smul",
"is_unit.sub_iff",
"mul_one",
"mul_smul_comm",
"one_mul",
"resolvent_set",
"smul_mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mem_iff {r : Rˣ} {a : Aˣ} :
(r : R) ∈ σ (a : A) ↔ (↑r⁻¹ : R) ∈ σ (↑a⁻¹ : A) | not_iff_not.2 $ ⟨inv_mem_resolvent_set, inv_mem_resolvent_set⟩ | lemma | spectrum.inv_mem_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"inv_mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem_resolvent_set_of_unit (a : Aˣ) : 0 ∈ resolvent_set R (a : A) | by simpa only [mem_resolvent_set_iff, ←not_mem_iff, zero_not_mem_iff] using a.is_unit | lemma | spectrum.zero_mem_resolvent_set_of_unit | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"resolvent_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_mem_of_unit {a : Aˣ} {r : R} (hr : r ∈ σ (a : A)) : r ≠ 0 | λ hn, (hn ▸ hr) (zero_mem_resolvent_set_of_unit a) | lemma | spectrum.ne_zero_of_mem_of_unit | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_iff {a : A} {r s : R} :
r + s ∈ σ a ↔ r ∈ σ (-↑ₐs + a) | by simp only [mem_iff, sub_neg_eq_add, ←sub_sub, map_add] | lemma | spectrum.add_mem_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem_add_iff {a : A} {r s : R} :
r + s ∈ σ (↑ₐs + a) ↔ r ∈ σ a | by rw [add_mem_iff, neg_add_cancel_left] | lemma | spectrum.add_mem_add_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_smul_iff {a : A} {s : R} {r : Rˣ} :
r • s ∈ σ (r • a) ↔ s ∈ σ a | by simp only [mem_iff, not_iff_not, algebra.algebra_map_eq_smul_one, smul_assoc, ←smul_sub,
is_unit_smul_iff] | lemma | spectrum.smul_mem_smul_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"is_unit_smul_iff",
"not_iff_not",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_smul_eq_smul (a : A) (r : Rˣ) :
σ (r • a) = r • σ a | begin
ext,
have x_eq : x = r • r⁻¹ • x, by simp,
nth_rewrite 0 x_eq,
rw smul_mem_smul_iff,
split,
{ exact λ h, ⟨r⁻¹ • x, ⟨h, by simp⟩⟩},
{ rintros ⟨_, _, x'_eq⟩, simpa [←x'_eq],}
end | theorem | spectrum.unit_smul_eq_smul | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_mem_mul_iff_mem_swap_mul {a b : A} {r : Rˣ} :
↑r ∈ σ (a * b) ↔ ↑r ∈ σ (b * a) | begin
have h₁ : ∀ x y : A, is_unit (1 - x * y) → is_unit (1 - y * x),
{ refine λ x y h, ⟨⟨1 - y * x, 1 + y * h.unit.inv * x, _, _⟩, rfl⟩,
calc (1 - y * x) * (1 + y * (is_unit.unit h).inv * x)
= (1 - y * x) + y * ((1 - x * y) * h.unit.inv) * x : by noncomm_ring
... = 1 : by simp only [units.inv_eq_co... | theorem | spectrum.unit_mem_mul_iff_mem_swap_mul | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"is_unit",
"is_unit.coe_inv_mul",
"is_unit.mul_coe_inv",
"is_unit.smul_sub_iff_sub_inv_smul",
"is_unit.unit",
"mul_one",
"not_iff_not",
"units.inv_eq_coe_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_units_mul_eq_swap_mul {a b : A} :
(coe : Rˣ → R) ⁻¹' σ (a * b) = coe ⁻¹' σ (b * a) | set.ext $ λ _, unit_mem_mul_iff_mem_swap_mul | theorem | spectrum.preimage_units_mul_eq_swap_mul | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mem_resolvent_set_iff {r : R} {a : A} :
star r ∈ resolvent_set R a ↔ r ∈ resolvent_set R (star a) | by refine ⟨λ h, _, λ h, _⟩;
simpa only [mem_resolvent_set_iff, algebra.algebra_map_eq_smul_one, star_sub, star_smul,
star_star, star_one] using is_unit.star h | lemma | spectrum.star_mem_resolvent_set_iff | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra.algebra_map_eq_smul_one",
"is_unit.star",
"resolvent_set",
"star_one",
"star_star",
"star_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_star (a : A) : σ (star a) = star (σ a) | by { ext, simpa only [set.mem_star, mem_iff, not_iff_not] using star_mem_resolvent_set_iff.symm } | lemma | spectrum.map_star | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"not_iff_not",
"set.mem_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_subalgebra {S : subalgebra R A} (a : S) : spectrum R (a : A) ⊆ spectrum R a | compl_subset_compl.2 (λ _, is_unit.map S.val) | lemma | spectrum.subset_subalgebra | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit.map",
"spectrum",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_star_subalgebra [star_ring R] [star_ring A] [star_module R A] {S : star_subalgebra R A}
(a : S) : spectrum R (a : A) ⊆ spectrum R a | compl_subset_compl.2 (λ _, is_unit.map S.subtype) | lemma | spectrum.subset_star_subalgebra | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit.map",
"spectrum",
"star_module",
"star_ring",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_add_eq (a : A) (r : R) : {r} + (σ a) = σ (↑ₐr + a) | ext $ λ x,
by rw [singleton_add, image_add_left, mem_preimage, add_comm, add_mem_iff, map_neg, neg_neg] | lemma | spectrum.singleton_add_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_singleton_eq (a : A) (r : R) : (σ a) + {r} = σ (a + ↑ₐr) | add_comm {r} (σ a) ▸ add_comm (algebra_map R A r) a ▸ singleton_add_eq a r | lemma | spectrum.add_singleton_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_eq (a : A) (r : R) : r +ᵥ (σ a) = σ (↑ₐr + a) | (singleton_add).symm.trans $ singleton_add_eq a r | lemma | spectrum.vadd_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq (a : A) : -(σ a) = σ (-a) | set.ext $ λ x, by simp only [mem_neg, mem_iff, map_neg, ←neg_add', is_unit.neg_iff, sub_neg_eq_add] | lemma | spectrum.neg_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"is_unit.neg_iff",
"neg_eq",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_sub_eq (a : A) (r : R) :
{r} - (σ a) = σ (↑ₐr - a) | by rw [sub_eq_add_neg, neg_eq, singleton_add_eq, sub_eq_add_neg] | lemma | spectrum.singleton_sub_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_singleton_eq (a : A) (r : R) :
(σ a) - {r} = σ (a - ↑ₐr) | by simpa only [neg_sub, neg_eq] using congr_arg has_neg.neg (singleton_sub_eq a r) | lemma | spectrum.sub_singleton_eq | algebra.algebra | src/algebra/algebra/spectrum.lean | [
"algebra.star.pointwise",
"algebra.star.subalgebra",
"tactic.noncomm_ring"
] | [
"neg_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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