statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
of_nonneg_iff [non_unital_ring R] [partial_order R] [star_ring R]
(h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y)
(h_nonneg_iff : ∀ x : R, 0 ≤ x ↔ x ∈ add_submonoid.closure (set.range $ λ s : R, star s * s)) :
star_ordered_ring R | { add_le_add_left := @h_add,
le_iff := λ x y,
begin
haveI : covariant_class R R (+) (≤) := ⟨λ _ _ _ h, h_add h _⟩,
simpa only [←sub_eq_iff_eq_add', sub_nonneg, exists_eq_right'] using h_nonneg_iff (y - x),
end,
.. ‹star_ring R› } | def | star_ordered_ring.of_nonneg_iff | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"covariant_class",
"exists_eq_right'",
"non_unital_ring",
"set.range",
"star_ordered_ring",
"star_ring"
] | When `R` is a non-unital ring, to construct a `star_ordered_ring` instance it suffices to
show that the nonnegative elements are precisely those elements in the `add_submonoid` generated
by `star s * s` for `s : R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_nonneg_iff' [non_unital_ring R] [partial_order R] [star_ring R]
(h_add : ∀ {x y : R}, x ≤ y → ∀ z, z + x ≤ z + y)
(h_nonneg_iff : ∀ x : R, 0 ≤ x ↔ ∃ s, x = star s * s) :
star_ordered_ring R | of_le_iff @h_add
begin
haveI : covariant_class R R (+) (≤) := ⟨λ _ _ _ h, h_add h _⟩,
simpa [sub_eq_iff_eq_add', sub_nonneg] using λ x y, h_nonneg_iff (y - x),
end | def | star_ordered_ring.of_nonneg_iff' | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"covariant_class",
"non_unital_ring",
"star_ordered_ring",
"star_ring"
] | When `R` is a non-unital ring, to construct a `star_ordered_ring` instance it suffices to
show that the nonnegative elements are precisely those elements of the form `star s * s`
for `s : R`.
This is provided for convenience because it holds in many common scenarios (e.g.,`ℝ`, `ℂ`, or
any C⋆-algebra), and obviates the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonneg_iff [non_unital_semiring R] [partial_order R] [star_ordered_ring R]
{x : R} : 0 ≤ x ↔ x ∈ add_submonoid.closure (set.range $ λ s : R, star s * s) | by simp only [le_iff, zero_add, exists_eq_right'] | lemma | star_ordered_ring.nonneg_iff | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"exists_eq_right'",
"non_unital_semiring",
"set.range",
"star_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self_nonneg (r : R) : 0 ≤ star r * r | star_ordered_ring.nonneg_iff.mpr $ add_submonoid.subset_closure ⟨r, rfl⟩ | lemma | star_mul_self_nonneg | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self_nonneg' (r : R) : 0 ≤ r * star r | by { nth_rewrite_rhs 0 [←star_star r], exact star_mul_self_nonneg (star r) } | lemma | star_mul_self_nonneg' | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"star_mul_self_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate_nonneg {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ star c * a * c | begin
rw star_ordered_ring.nonneg_iff at ha,
refine add_submonoid.closure_induction ha (λ x hx, _) (by rw [mul_zero, zero_mul])
(λ x y hx hy, _),
{ obtain ⟨x, rfl⟩ := hx,
convert star_mul_self_nonneg (x * c) using 1,
rw [star_mul, ←mul_assoc, mul_assoc _ _ c] },
{ calc 0 ≤ star c * x * c + 0 : by rw... | lemma | conjugate_nonneg | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"mul_assoc",
"mul_zero",
"star_mul_self_nonneg",
"star_ordered_ring.nonneg_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate_nonneg' {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ c * a * star c | by simpa only [star_star] using conjugate_nonneg ha (star c) | lemma | conjugate_nonneg' | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"conjugate_nonneg",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate_le_conjugate {a b : R} (hab : a ≤ b) (c : R) : star c * a * c ≤ star c * b * c | begin
rw [star_ordered_ring.le_iff] at hab ⊢,
obtain ⟨p, hp, rfl⟩ := hab,
simp_rw [←star_ordered_ring.nonneg_iff] at hp ⊢,
exact ⟨star c * p * c, conjugate_nonneg hp c, by simp only [add_mul, mul_add]⟩,
end | lemma | conjugate_le_conjugate | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"conjugate_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate_le_conjugate' {a b : R} (hab : a ≤ b) (c : R) : c * a * star c ≤ c * b * star c | by simpa only [star_star] using conjugate_le_conjugate hab (star c) | lemma | conjugate_le_conjugate' | algebra.star | src/algebra/star/order.lean | [
"algebra.star.basic",
"group_theory.submonoid.basic"
] | [
"conjugate_le_conjugate",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_apply [Π i, has_star (f i)] (x : Π i, f i) (i : I) : star x i = star (x i) | rfl | lemma | pi.star_apply | algebra.star | src/algebra/star/pi.lean | [
"algebra.star.basic",
"algebra.ring.pi"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_def [Π i, has_star (f i)] (x : Π i, f i) : star x = λ i, star (x i) | rfl | lemma | pi.star_def | algebra.star | src/algebra/star/pi.lean | [
"algebra.star.basic",
"algebra.ring.pi"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
single_star [Π i, add_monoid (f i)] [Π i, star_add_monoid (f i)] [decidable_eq I]
(i : I) (a : f i) :
pi.single i (star a) = star (pi.single i a) | single_op (λ i, @star (f i) _) (λ i, star_zero _) i a | lemma | pi.single_star | algebra.star | src/algebra/star/pi.lean | [
"algebra.star.basic",
"algebra.ring.pi"
] | [
"add_monoid",
"star_add_monoid",
"star_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_star [Π i, has_star (f i)] [decidable_eq I]
(h : Π (i : I), f i) (i : I) (a : f i) :
function.update (star h) i (star a) = star (function.update h i a) | funext $ λ j, (apply_update (λ i, star) h i a j).symm | lemma | function.update_star | algebra.star | src/algebra/star/pi.lean | [
"algebra.star.basic",
"algebra.ring.pi"
] | [
"apply_update",
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_sum_elim {I J α : Type*} (x : I → α) (y : J → α) [has_star α] :
star (sum.elim x y) = sum.elim (star x) (star y) | by { ext x, cases x; simp } | lemma | function.star_sum_elim | algebra.star | src/algebra/star/pi.lean | [
"algebra.star.basic",
"algebra.ring.pi"
] | [
"has_star",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_star [has_star α] : has_star (set α) | ⟨preimage has_star.star⟩ | def | set.has_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | The set `(star s : set α)` is defined as `{x | star x ∈ s}` in locale `pointwise`.
In the usual case where `star` is involutive, it is equal to `{star s | x ∈ s}`, see
`set.image_star`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_empty [has_star α] : (∅ : set α)⋆ = ∅ | rfl | lemma | set.star_empty | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_univ [has_star α] : (univ : set α)⋆ = univ | rfl | lemma | set.star_univ | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_star [has_involutive_star α] {s : set α} : (s⋆).nonempty ↔ s.nonempty | star_involutive.surjective.nonempty_preimage | lemma | set.nonempty_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty.star [has_involutive_star α] {s : set α} (h : s.nonempty) :
(s⋆).nonempty | nonempty_star.2 h | lemma | set.nonempty.star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_star [has_star α] : a ∈ s⋆ ↔ a⋆ ∈ s | iff.rfl | lemma | set.mem_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mem_star [has_involutive_star α] : a⋆ ∈ s⋆ ↔ a ∈ s | by simp only [mem_star, star_star] | lemma | set.star_mem_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_preimage [has_star α] : has_star.star ⁻¹' s = s⋆ | rfl | lemma | set.star_preimage | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_star [has_involutive_star α] : has_star.star '' s = s⋆ | by { simp only [← star_preimage], rw [image_eq_preimage_of_inverse]; intro; simp only [star_star] } | lemma | set.image_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_star [has_star α] : (s ∩ t)⋆ = s⋆ ∩ t⋆ | preimage_inter | lemma | set.inter_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_star [has_star α] : (s ∪ t)⋆ = s⋆ ∪ t⋆ | preimage_union | lemma | set.union_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inter_star {ι : Sort*} [has_star α] (s : ι → set α) : (⋂ i, s i)⋆ = ⋂ i, (s i)⋆ | preimage_Inter | lemma | set.Inter_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_star {ι : Sort*} [has_star α] (s : ι → set α) : (⋃ i, s i)⋆ = ⋃ i, (s i)⋆ | preimage_Union | lemma | set.Union_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_star [has_star α] : (sᶜ)⋆ = (s⋆)ᶜ | preimage_compl | lemma | set.compl_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_subset_star [has_involutive_star α] {s t : set α} : s⋆ ⊆ t⋆ ↔ s ⊆ t | equiv.star.surjective.preimage_subset_preimage_iff | lemma | set.star_subset_star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_subset [has_involutive_star α] {s t : set α} : s⋆ ⊆ t ↔ s ⊆ t⋆ | by { rw [← star_subset_star, star_star] } | lemma | set.star_subset | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.star [has_involutive_star α] {s : set α} (hs : s.finite) : s⋆.finite | hs.preimage $ star_injective.inj_on _ | lemma | set.finite.star | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_singleton {β : Type*} [has_involutive_star β] (x : β) : ({x} : set β)⋆ = {x⋆} | by { ext1 y, rw [mem_star, mem_singleton_iff, mem_singleton_iff, star_eq_iff_star_eq, eq_comm], } | lemma | set.star_singleton | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"has_involutive_star",
"star_eq_iff_star_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul [monoid α] [star_semigroup α] (s t : set α) :
(s * t)⋆ = t⋆ * s⋆ | by simp_rw [←image_star, ←image2_mul, image_image2, image2_image_left, image2_image_right,
star_mul, image2_swap _ s t] | lemma | set.star_mul | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"monoid",
"star_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_add [add_monoid α] [star_add_monoid α] (s t : set α) :
(s + t)⋆ = s⋆ + t⋆ | by simp_rw [←image_star, ←image2_add, image_image2, image2_image_left, image2_image_right, star_add] | lemma | set.star_add | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"add_monoid",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_inv [group α] [star_semigroup α] (s : set α) : (s⁻¹)⋆ = (s⋆)⁻¹ | by { ext, simp only [mem_star, mem_inv, star_inv] } | lemma | set.star_inv | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"group",
"star_inv",
"star_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_inv' [division_semiring α] [star_ring α] (s : set α) : (s⁻¹)⋆ = (s⋆)⁻¹ | by { ext, simp only [mem_star, mem_inv, star_inv'] } | lemma | set.star_inv' | algebra.star | src/algebra/star/pointwise.lean | [
"algebra.star.basic",
"data.set.finite",
"data.set.pointwise.basic"
] | [
"division_semiring",
"star_inv'",
"star_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_star [has_star R] [has_star S] (x : R × S) : (star x).1 = star x.1 | rfl | lemma | prod.fst_star | algebra.star | src/algebra/star/prod.lean | [
"algebra.star.basic",
"algebra.ring.prod",
"algebra.module.prod"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_star [has_star R] [has_star S] (x : R × S) : (star x).2 = star x.2 | rfl | lemma | prod.snd_star | algebra.star | src/algebra/star/prod.lean | [
"algebra.star.basic",
"algebra.ring.prod",
"algebra.module.prod"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_def [has_star R] [has_star S] (x : R × S) : star x = (star x.1, star x.2) | rfl | lemma | prod.star_def | algebra.star | src/algebra/star/prod.lean | [
"algebra.star.basic",
"algebra.ring.prod",
"algebra.module.prod"
] | [
"has_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.embed_product_star [monoid R] [star_semigroup R] (u : Rˣ) :
units.embed_product R (star u) = star (units.embed_product R u) | rfl | lemma | units.embed_product_star | algebra.star | src/algebra/star/prod.lean | [
"algebra.star.basic",
"algebra.ring.prod",
"algebra.module.prod"
] | [
"monoid",
"star_semigroup",
"units.embed_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint [has_star R] (x : R) : Prop | star x = x | def | is_self_adjoint | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_star"
] | An element is self-adjoint if it is equal to its star. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_star_normal [has_mul R] [has_star R] (x : R) : Prop | (star_comm_self : commute (star x) x) | class | is_star_normal | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"commute",
"has_star"
] | An element of a star monoid is normal if it commutes with its adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_comm_self' [has_mul R] [has_star R] (x : R) [is_star_normal x] :
(star x) * x = x * star x | is_star_normal.star_comm_self | lemma | star_comm_self' | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_star",
"is_star_normal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
all [has_star R] [has_trivial_star R] (r : R) : is_self_adjoint r | star_trivial _ | lemma | is_self_adjoint.all | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_star",
"has_trivial_star",
"is_self_adjoint"
] | All elements are self-adjoint when `star` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_eq [has_star R] {x : R} (hx : is_self_adjoint x) : star x = x | hx | lemma | is_self_adjoint.star_eq | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_star",
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_iff [has_star R] {x : R} : is_self_adjoint x ↔ star x = x | iff.rfl | lemma | is_self_adjoint_iff | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_star",
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_iff [has_involutive_star R] {x : R} : is_self_adjoint (star x) ↔ is_self_adjoint x | by simpa only [is_self_adjoint, star_star] using eq_comm | lemma | is_self_adjoint.star_iff | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_involutive_star",
"is_self_adjoint",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self [semigroup R] [star_semigroup R] (x : R) : is_self_adjoint (star x * x) | by simp only [is_self_adjoint, star_mul, star_star] | lemma | is_self_adjoint.star_mul_self | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"semigroup",
"star_semigroup",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_self [semigroup R] [star_semigroup R] (x : R) : is_self_adjoint (x * star x) | by simpa only [star_star] using star_mul_self (star x) | lemma | is_self_adjoint.mul_star_self | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"semigroup",
"star_semigroup",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_hom_apply {F R S : Type*} [has_star R] [has_star S] [star_hom_class F R S]
{x : R} (hx : is_self_adjoint x) (f : F) : is_self_adjoint (f x) | show star (f x) = f x, from map_star f x ▸ congr_arg f hx | lemma | is_self_adjoint.star_hom_apply | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_star",
"is_self_adjoint",
"star_hom_class"
] | Functions in a `star_hom_class` preserve self-adjoint elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_self_adjoint_zero : is_self_adjoint (0 : R) | star_zero R | lemma | is_self_adjoint_zero | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"star_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add {x y : R} (hx : is_self_adjoint x) (hy : is_self_adjoint y) : is_self_adjoint (x + y) | by simp only [is_self_adjoint_iff, star_add, hx.star_eq, hy.star_eq] | lemma | is_self_adjoint.add | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0 {x : R} (hx : is_self_adjoint x) : is_self_adjoint (bit0 x) | by simp only [is_self_adjoint_iff, star_bit0, hx.star_eq] | lemma | is_self_adjoint.bit0 | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg {x : R} (hx : is_self_adjoint x) : is_self_adjoint (-x) | by simp only [is_self_adjoint_iff, star_neg, hx.star_eq] | lemma | is_self_adjoint.neg | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub {x y : R} (hx : is_self_adjoint x) (hy : is_self_adjoint y) : is_self_adjoint (x - y) | by simp only [is_self_adjoint_iff, star_sub, hx.star_eq, hy.star_eq] | lemma | is_self_adjoint.sub | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_add_star_self (x : R) : is_self_adjoint (x + star x) | by simp only [is_self_adjoint_iff, add_comm, star_add, star_star] | lemma | is_self_adjoint_add_star_self | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_star_add_self (x : R) : is_self_adjoint (star x + x) | by simp only [is_self_adjoint_iff, add_comm, star_add, star_star] | lemma | is_self_adjoint_star_add_self | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate {x : R} (hx : is_self_adjoint x) (z : R) : is_self_adjoint (z * x * star z) | by simp only [is_self_adjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq] | lemma | is_self_adjoint.conjugate | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"mul_assoc",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate' {x : R} (hx : is_self_adjoint x) (z : R) : is_self_adjoint (star z * x * z) | by simp only [is_self_adjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq] | lemma | is_self_adjoint.conjugate' | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"mul_assoc",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_star_normal {x : R} (hx : is_self_adjoint x) : is_star_normal x | ⟨by simp only [hx.star_eq]⟩ | lemma | is_self_adjoint.is_star_normal | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_star_normal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_one : is_self_adjoint (1 : R) | star_one R | lemma | is_self_adjoint_one | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"star_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow {x : R} (hx : is_self_adjoint x) (n : ℕ) : is_self_adjoint (x ^ n) | by simp only [is_self_adjoint_iff, star_pow, hx.star_eq] | lemma | is_self_adjoint.pow | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1 {x : R} (hx : is_self_adjoint x) : is_self_adjoint (bit1 x) | by simp only [is_self_adjoint_iff, star_bit1, hx.star_eq] | lemma | is_self_adjoint.bit1 | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_bit1"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_nat_cast (n : ℕ) : is_self_adjoint (n : R) | star_nat_cast _ | lemma | is_self_adjoint_nat_cast | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"star_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul {x y : R} (hx : is_self_adjoint x) (hy : is_self_adjoint y) : is_self_adjoint (x * y) | by simp only [is_self_adjoint_iff, star_mul', hx.star_eq, hy.star_eq] | lemma | is_self_adjoint.mul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_int_cast (z : ℤ) : is_self_adjoint (z : R) | star_int_cast _ | lemma | is_self_adjoint_int_cast | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"star_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv {x : R} (hx : is_self_adjoint x) : is_self_adjoint x⁻¹ | by simp only [is_self_adjoint_iff, star_inv', hx.star_eq] | lemma | is_self_adjoint.inv | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow {x : R} (hx : is_self_adjoint x) (n : ℤ) : is_self_adjoint (x ^ n) | by simp only [is_self_adjoint_iff, star_zpow₀, hx.star_eq] | lemma | is_self_adjoint.zpow | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_zpow₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_self_adjoint_rat_cast (x : ℚ) : is_self_adjoint (x : R) | star_rat_cast _ | lemma | is_self_adjoint_rat_cast | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"star_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div {x y : R} (hx : is_self_adjoint x) (hy : is_self_adjoint y) : is_self_adjoint (x / y) | by simp only [is_self_adjoint_iff, star_div', hx.star_eq, hy.star_eq] | lemma | is_self_adjoint.div | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff",
"star_div'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul {r : R} (hr : is_self_adjoint r) {x : A} (hx : is_self_adjoint x) :
is_self_adjoint (r • x) | by simp only [is_self_adjoint_iff, star_smul, hr.star_eq, hx.star_eq] | lemma | is_self_adjoint.smul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_self_adjoint",
"is_self_adjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_adjoint [add_group R] [star_add_monoid R] : add_subgroup R | { carrier := {x | is_self_adjoint x},
zero_mem' := star_zero R,
add_mem' := λ _ _ hx, hx.add,
neg_mem' := λ _ hx, hx.neg } | def | self_adjoint | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"add_group",
"add_subgroup",
"is_self_adjoint",
"star_add_monoid",
"star_zero"
] | The self-adjoint elements of a star additive group, as an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
skew_adjoint [add_comm_group R] [star_add_monoid R] : add_subgroup R | { carrier := {x | star x = -x},
zero_mem' := show star (0 : R) = -0, by simp only [star_zero, neg_zero],
add_mem' := λ x y (hx : star x = -x) (hy : star y = -y),
show star (x + y) = -(x + y), by rw [star_add x y, hx, hy, neg_add],
neg_mem' := λ x (hx : star x = -x), show star (-x) = (- -x), by sim... | def | skew_adjoint | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"add_comm_group",
"add_subgroup",
"star_add_monoid",
"star_neg",
"star_zero"
] | The skew-adjoint elements of a star additive group, as an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_iff {x : R} : x ∈ self_adjoint R ↔ star x = x | by { rw [←add_subgroup.mem_carrier], exact iff.rfl } | lemma | self_adjoint.mem_iff | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_coe_eq {x : self_adjoint R} : star (x : R) = x | x.prop | lemma | self_adjoint.star_coe_eq | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ↑(1 : self_adjoint R) = (1 : R) | rfl | lemma | self_adjoint.coe_one | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (x : self_adjoint R) (n : ℕ) : ↑(x ^ n) = (x : R) ^ n | rfl | lemma | self_adjoint.coe_pow | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (x y : self_adjoint R) : ↑(x * y) = (x : R) * y | rfl | lemma | self_adjoint.coe_mul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (x : self_adjoint R) : ↑(x⁻¹) = (x : R)⁻¹ | rfl | lemma | self_adjoint.coe_inv | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (x y : self_adjoint R) : ↑(x / y) = (x / y : R) | rfl | lemma | self_adjoint.coe_div | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow (x : self_adjoint R) (z : ℤ) : ↑(x ^ z) = (x : R) ^ z | rfl | lemma | self_adjoint.coe_zpow | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rat_cast (x : ℚ) : ↑(x : self_adjoint R) = (x : R) | rfl | lemma | self_adjoint.coe_rat_cast | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_qsmul : has_smul ℚ (self_adjoint R) | ⟨λ a x, ⟨a • x, by rw rat.smul_def; exact is_self_adjoint.mul (is_self_adjoint_rat_cast a) x.prop⟩⟩ | instance | self_adjoint.has_qsmul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_smul",
"is_self_adjoint.mul",
"is_self_adjoint_rat_cast",
"rat.smul_def",
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rat_smul (x : self_adjoint R) (a : ℚ) : ↑(a • x) = a • (x : R) | rfl | lemma | self_adjoint.coe_rat_smul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [has_smul R A] [star_module R A] (r : R) (x : self_adjoint A) :
↑(r • x) = r • (x : A) | rfl | lemma | self_adjoint.coe_smul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_smul",
"self_adjoint",
"star_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff {x : R} : x ∈ skew_adjoint R ↔ star x = -x | by { rw [←add_subgroup.mem_carrier], exact iff.rfl } | lemma | skew_adjoint.mem_iff | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"skew_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_coe_eq {x : skew_adjoint R} : star (x : R) = -x | x.prop | lemma | skew_adjoint.star_coe_eq | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"skew_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_mem {x : R} (hx : x ∈ skew_adjoint R) : bit0 x ∈ skew_adjoint R | by rw [mem_iff, star_bit0, mem_iff.mp hx, bit0, bit0, neg_add] | lemma | skew_adjoint.bit0_mem | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"skew_adjoint",
"star_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate {x : R} (hx : x ∈ skew_adjoint R) (z : R) : z * x * star z ∈ skew_adjoint R | by simp only [mem_iff, star_mul, star_star, mem_iff.mp hx, neg_mul, mul_neg, mul_assoc] | lemma | skew_adjoint.conjugate | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"mul_assoc",
"mul_neg",
"neg_mul",
"skew_adjoint",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conjugate' {x : R} (hx : x ∈ skew_adjoint R) (z : R) : star z * x * z ∈ skew_adjoint R | by simp only [mem_iff, star_mul, star_star, mem_iff.mp hx, neg_mul, mul_neg, mul_assoc] | lemma | skew_adjoint.conjugate' | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"mul_assoc",
"mul_neg",
"neg_mul",
"skew_adjoint",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_star_normal_of_mem {x : R} (hx : x ∈ skew_adjoint R) : is_star_normal x | ⟨by { simp only [mem_iff] at hx, simp only [hx, commute.neg_left] }⟩ | lemma | skew_adjoint.is_star_normal_of_mem | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"commute.neg_left",
"is_star_normal",
"skew_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem [monoid R] [distrib_mul_action R A] [star_module R A] (r : R) {x : A}
(h : x ∈ skew_adjoint A) : r • x ∈ skew_adjoint A | by rw [mem_iff, star_smul, star_trivial, mem_iff.mp h, smul_neg r] | lemma | skew_adjoint.smul_mem | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"distrib_mul_action",
"monoid",
"skew_adjoint",
"smul_neg",
"star_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [monoid R] [distrib_mul_action R A] [star_module R A]
(r : R) (x : skew_adjoint A) : ↑(r • x) = r • (x : A) | rfl | lemma | skew_adjoint.coe_smul | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"distrib_mul_action",
"monoid",
"skew_adjoint",
"star_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_self_adjoint.smul_mem_skew_adjoint [ring R] [add_comm_group A] [module R A]
[star_add_monoid R] [star_add_monoid A] [star_module R A] {r : R}
(hr : r ∈ skew_adjoint R) {a : A} (ha : is_self_adjoint a) :
r • a ∈ skew_adjoint A | (star_smul _ _).trans $ (congr_arg2 _ hr ha).trans $ neg_smul _ _ | lemma | is_self_adjoint.smul_mem_skew_adjoint | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"add_comm_group",
"congr_arg2",
"is_self_adjoint",
"module",
"neg_smul",
"ring",
"skew_adjoint",
"star_add_monoid",
"star_module"
] | Scalar multiplication of a self-adjoint element by a skew-adjoint element produces a
skew-adjoint element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_self_adjoint_smul_of_mem_skew_adjoint [ring R] [add_comm_group A] [module R A]
[star_add_monoid R] [star_add_monoid A] [star_module R A] {r : R}
(hr : r ∈ skew_adjoint R) {a : A} (ha : a ∈ skew_adjoint A) :
is_self_adjoint (r • a) | (star_smul _ _).trans $ (congr_arg2 _ hr ha).trans $ neg_smul_neg _ _ | lemma | is_self_adjoint_smul_of_mem_skew_adjoint | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"add_comm_group",
"congr_arg2",
"is_self_adjoint",
"module",
"neg_smul_neg",
"ring",
"skew_adjoint",
"star_add_monoid",
"star_module"
] | Scalar multiplication of a skew-adjoint element by a skew-adjoint element produces a
self-adjoint element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_star_normal_zero [semiring R] [star_ring R] : is_star_normal (0 : R) | ⟨by simp only [star_comm_self, star_zero]⟩ | instance | is_star_normal_zero | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_star_normal",
"semiring",
"star_ring",
"star_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_star_normal_one [monoid R] [star_semigroup R] : is_star_normal (1 : R) | ⟨by simp only [star_comm_self, star_one]⟩ | instance | is_star_normal_one | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_star_normal",
"monoid",
"star_one",
"star_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_star_normal_star_self [monoid R] [star_semigroup R] {x : R} [is_star_normal x] :
is_star_normal (star x) | ⟨show star (star x) * (star x) = (star x) * star (star x), by rw [star_star, star_comm_self']⟩ | instance | is_star_normal_star_self | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"is_star_normal",
"monoid",
"star_comm_self'",
"star_semigroup",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_trivial_star.is_star_normal [monoid R] [star_semigroup R]
[has_trivial_star R] {x : R} : is_star_normal x | ⟨by rw [star_trivial]⟩ | instance | has_trivial_star.is_star_normal | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"has_trivial_star",
"is_star_normal",
"monoid",
"star_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid.is_star_normal [comm_monoid R] [star_semigroup R] {x : R} :
is_star_normal x | ⟨mul_comm _ _⟩ | instance | comm_monoid.is_star_normal | algebra.star | src/algebra/star/self_adjoint.lean | [
"algebra.star.basic",
"group_theory.subgroup.basic"
] | [
"comm_monoid",
"is_star_normal",
"star_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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