fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
isAlt_iff_eq_neg_flip [NoZeroDivisors R] [CharZero R] {B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R} :
B.IsAlt ↔ B = -B.flip := by
constructor <;> intro h
· ext
simp_rw [neg_apply, flip_apply]
exact (h.neg _ _).symm
intro x
let h' := congr_fun₂ h x x
simp only [neg_apply, flip_apply, ← add_eq_zero_iff_eq_neg] at h'
... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isAlt_iff_eq_neg_flip | null |
orthogonalBilin (N : Submodule R₁ M₁) (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M) : Submodule R₁ M₁ where
carrier := { m | ∀ n ∈ N, B.IsOrtho n m }
zero_mem' x _ := B.isOrtho_zero_right x
add_mem' hx hy n hn := by
rw [LinearMap.IsOrtho, map_add, show B n _ = 0 from hx n hn, show B n _ = 0 from hy n hn,
zero_add]
sm... | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | orthogonalBilin | The orthogonal complement of a submodule `N` with respect to some bilinear map is the set of
elements `x` which are orthogonal to all elements of `N`; i.e., for all `y` in `N`, `B x y = 0`.
Note that for general (neither symmetric nor antisymmetric) bilinear maps this definition has a
chirality; in addition to this "l... |
mem_orthogonalBilin_iff {m : M₁} : m ∈ N.orthogonalBilin B ↔ ∀ n ∈ N, B.IsOrtho n m :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | mem_orthogonalBilin_iff | null |
orthogonalBilin_le (h : N ≤ L) : L.orthogonalBilin B ≤ N.orthogonalBilin B :=
fun _ hn l hl ↦ hn l (h hl) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | orthogonalBilin_le | null |
le_orthogonalBilin_orthogonalBilin (b : B.IsRefl) :
N ≤ (N.orthogonalBilin B).orthogonalBilin B := fun n hn _m hm ↦ b _ _ (hm n hn) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | le_orthogonalBilin_orthogonalBilin | null |
span_singleton_inf_orthogonal_eq_bot (B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂) (x : V₁)
(hx : ¬B.IsOrtho x x) : (K₁ ∙ x) ⊓ Submodule.orthogonalBilin (K₁ ∙ x) B = ⊥ := by
rw [← Finset.coe_singleton]
refine eq_bot_iff.2 fun y h ↦ ?_
obtain ⟨μ, -, rfl⟩ := Submodule.mem_span_finset.1 h.1
replace h := h.2 x (by simp [Subm... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | span_singleton_inf_orthogonal_eq_bot | null |
orthogonal_span_singleton_eq_to_lin_ker {B : V →ₗ[K] V →ₛₗ[J] V₂} (x : V) :
Submodule.orthogonalBilin (K ∙ x) B = LinearMap.ker (B x) := by
ext y
simp_rw [Submodule.mem_orthogonalBilin_iff, LinearMap.mem_ker, Submodule.mem_span_singleton]
constructor
· exact fun h ↦ h x ⟨1, one_smul _ _⟩
· rintro h _ ⟨z, ... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | orthogonal_span_singleton_eq_to_lin_ker | null |
span_singleton_sup_orthogonal_eq_top {B : V →ₗ[K] V →ₗ[K] K} {x : V} (hx : ¬B.IsOrtho x x) :
(K ∙ x) ⊔ Submodule.orthogonalBilin (N := K ∙ x) (B := B) = ⊤ := by
rw [orthogonal_span_singleton_eq_to_lin_ker]
exact (B x).span_singleton_sup_ker_eq_top hx | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | span_singleton_sup_orthogonal_eq_top | null |
isCompl_span_singleton_orthogonal {B : V →ₗ[K] V →ₗ[K] K} {x : V} (hx : ¬B.IsOrtho x x) :
IsCompl (K ∙ x) (Submodule.orthogonalBilin (N := K ∙ x) (B := B)) :=
{ disjoint := disjoint_iff.2 <| span_singleton_inf_orthogonal_eq_bot B x hx
codisjoint := codisjoint_iff.2 <| span_singleton_sup_orthogonal_eq_top hx } | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isCompl_span_singleton_orthogonal | Given a bilinear form `B` and some `x` such that `B x x ≠ 0`, the span of the singleton of `x`
is complement to its orthogonal complement. |
IsAdjointPair (f : M → M₁) (g : M₁ → M) :=
∀ x y, B' (f x) y = B x (g y)
variable {B B' f g} | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAdjointPair | Given a pair of modules equipped with bilinear maps, this is the condition for a pair of
maps between them to be mutually adjoint. |
isAdjointPair_iff_comp_eq_compl₂ : IsAdjointPair B B' f g ↔ B'.comp f = B.compl₂ g := by
constructor <;> intro h
· ext x y
rw [comp_apply, compl₂_apply]
exact h x y
· intro _ _
rw [← compl₂_apply, ← comp_apply, h] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isAdjointPair_iff_comp_eq_compl₂ | null |
isAdjointPair_zero : IsAdjointPair B B' 0 0 := fun _ _ ↦ by
simp only [Pi.zero_apply, map_zero, zero_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isAdjointPair_zero | null |
isAdjointPair_id : IsAdjointPair B B (_root_.id : M → M) (_root_.id : M → M) :=
fun _ _ ↦ rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isAdjointPair_id | null |
isAdjointPair_one : IsAdjointPair B B (1 : Module.End R M) (1 : Module.End R M) :=
isAdjointPair_id | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isAdjointPair_one | null |
IsAdjointPair.add {f f' : M → M₁} {g g' : M₁ → M} (h : IsAdjointPair B B' f g)
(h' : IsAdjointPair B B' f' g') :
IsAdjointPair B B' (f + f') (g + g') := fun x _ ↦ by
rw [Pi.add_apply, Pi.add_apply, B'.map_add₂, (B x).map_add, h, h'] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAdjointPair.add | null |
IsAdjointPair.comp {f : M → M₁} {g : M₁ → M} {f' : M₁ → M₂} {g' : M₂ → M₁}
(h : IsAdjointPair B B' f g) (h' : IsAdjointPair B' B'' f' g') :
IsAdjointPair B B'' (f' ∘ f) (g ∘ g') := fun _ _ ↦ by
rw [Function.comp_def, Function.comp_def, h', h] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAdjointPair.comp | null |
IsAdjointPair.mul {f g f' g' : Module.End R M} (h : IsAdjointPair B B f g)
(h' : IsAdjointPair B B f' g') : IsAdjointPair B B (f * f') (g' * g) :=
h'.comp h | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAdjointPair.mul | null |
IsAdjointPair.sub (h : IsAdjointPair B B' f g) (h' : IsAdjointPair B B' f' g') :
IsAdjointPair B B' (f - f') (g - g') := fun x _ ↦ by
rw [Pi.sub_apply, Pi.sub_apply, B'.map_sub₂, (B x).map_sub, h, h'] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAdjointPair.sub | null |
IsAdjointPair.smul (c : R) (h : IsAdjointPair B B' f g) :
IsAdjointPair B B' (c • f) (c • g) := fun _ _ ↦ by
simp [h _] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAdjointPair.smul | null |
IsOrthogonal : Prop :=
∀ x y, B (f x) (f y) = B x y
variable {B f}
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthogonal | A linear transformation `f` is orthogonal with respect to a bilinear form `B` if `B` is
bi-invariant with respect to `f`. |
_root_.LinearEquiv.isAdjointPair_symm_iff {f : M ≃ M} :
LinearMap.IsAdjointPair B B f f.symm ↔ B.IsOrthogonal f :=
⟨fun hf x y ↦ by simpa using hf x (f y), fun hf x y ↦ by simpa using hf x (f.symm y)⟩ | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | _root_.LinearEquiv.isAdjointPair_symm_iff | null |
isOrthogonal_of_forall_apply_same {F : Type*} [FunLike F M M] [LinearMapClass F R M M]
(f : F) (h : IsLeftRegular (2 : R)) (hB : B.IsSymm) (hf : ∀ x, B (f x) (f x) = B x x) :
B.IsOrthogonal f := by
intro x y
suffices 2 * B (f x) (f y) = 2 * B x y from h this
have := hf (x + y)
simp only [map_add, Linear... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrthogonal_of_forall_apply_same | null |
IsPairSelfAdjoint (f : M → M) :=
IsAdjointPair B F f f | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsPairSelfAdjoint | The condition for an endomorphism to be "self-adjoint" with respect to a pair of bilinear maps
on the underlying module. In the case that these two maps are identical, this is the usual concept
of self adjointness. In the case that one of the maps is the negation of the other, this is the
usual concept of skew adjointn... |
protected IsSelfAdjoint (f : M → M) :=
IsAdjointPair B B f f | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsSelfAdjoint | An endomorphism of a module is self-adjoint with respect to a bilinear map if it serves as an
adjoint for itself. |
isPairSelfAdjointSubmodule : Submodule R (Module.End R M) where
carrier := { f | IsPairSelfAdjoint B F f }
zero_mem' := isAdjointPair_zero
add_mem' hf hg := hf.add hg
smul_mem' c _ h := h.smul c | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isPairSelfAdjointSubmodule | The set of pair-self-adjoint endomorphisms are a submodule of the type of all endomorphisms. |
IsSkewAdjoint (f : M → M) :=
IsAdjointPair B B f (-f) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsSkewAdjoint | An endomorphism of a module is skew-adjoint with respect to a bilinear map if its negation
serves as an adjoint. |
selfAdjointSubmodule :=
isPairSelfAdjointSubmodule B B | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | selfAdjointSubmodule | The set of self-adjoint endomorphisms of a module with bilinear map is a submodule. (In fact
it is a Jordan subalgebra.) |
skewAdjointSubmodule :=
isPairSelfAdjointSubmodule (-B) B
variable {B F}
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | skewAdjointSubmodule | The set of skew-adjoint endomorphisms of a module with bilinear map is a submodule. (In fact
it is a Lie subalgebra.) |
mem_isPairSelfAdjointSubmodule (f : Module.End R M) :
f ∈ isPairSelfAdjointSubmodule B F ↔ IsPairSelfAdjoint B F f :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | mem_isPairSelfAdjointSubmodule | null |
isPairSelfAdjoint_equiv (e : M₁ ≃ₗ[R] M) (f : Module.End R M) :
IsPairSelfAdjoint B F f ↔
IsPairSelfAdjoint (B.compl₁₂ e e) (F.compl₁₂ e e) (e.symm.conj f) := by
have hₗ :
(F.compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M)).comp (e.symm.conj f) =
(F.comp f).compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M) ... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isPairSelfAdjoint_equiv | null |
isSkewAdjoint_iff_neg_self_adjoint (f : M → M) :
B.IsSkewAdjoint f ↔ IsAdjointPair (-B) B f f :=
show (∀ x y, B (f x) y = B x ((-f) y)) ↔ ∀ x y, B (f x) y = (-B) x (f y) by simp
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isSkewAdjoint_iff_neg_self_adjoint | null |
mem_selfAdjointSubmodule (f : Module.End R M) :
f ∈ B.selfAdjointSubmodule ↔ B.IsSelfAdjoint f :=
Iff.rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | mem_selfAdjointSubmodule | null |
mem_skewAdjointSubmodule (f : Module.End R M) :
f ∈ B.skewAdjointSubmodule ↔ B.IsSkewAdjoint f := by
rw [isSkewAdjoint_iff_neg_self_adjoint]
exact Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | mem_skewAdjointSubmodule | null |
SeparatingLeft (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) : Prop :=
∀ x : M₁, (∀ y : M₂, B x y = 0) → x = 0
variable (M₁ M₂ I₁ I₂) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | SeparatingLeft | A bilinear map is called left-separating if
the only element that is left-orthogonal to every other element is `0`; i.e.,
for every nonzero `x` in `M₁`, there exists `y` in `M₂` with `B x y ≠ 0`. |
not_separatingLeft_zero [Nontrivial M₁] : ¬(0 : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M).SeparatingLeft :=
let ⟨m, hm⟩ := exists_ne (0 : M₁)
fun h ↦ hm (h m fun _n ↦ rfl)
variable {M₁ M₂ I₁ I₂} | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | not_separatingLeft_zero | In a non-trivial module, zero is not non-degenerate. |
SeparatingLeft.ne_zero [Nontrivial M₁] {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M}
(h : B.SeparatingLeft) : B ≠ 0 := fun h0 ↦ not_separatingLeft_zero M₁ M₂ I₁ I₂ <| h0 ▸ h | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | SeparatingLeft.ne_zero | null |
SeparatingLeft.congr (h : B.SeparatingLeft) :
(e₁.arrowCongr (e₂.arrowCongr (LinearEquiv.refl R M)) B).SeparatingLeft := by
intro x hx
rw [← e₁.symm.map_eq_zero_iff]
refine h (e₁.symm x) fun y ↦ ?_
specialize hx (e₂ y)
simp only [LinearEquiv.arrowCongr_apply, LinearEquiv.symm_apply_apply,
LinearEquiv.... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | SeparatingLeft.congr | null |
separatingLeft_congr_iff :
(e₁.arrowCongr (e₂.arrowCongr (LinearEquiv.refl R M)) B).SeparatingLeft ↔ B.SeparatingLeft :=
⟨fun h ↦ by
convert h.congr e₁.symm e₂.symm
ext x y
simp,
SeparatingLeft.congr e₁ e₂⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | separatingLeft_congr_iff | null |
SeparatingRight (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) : Prop :=
∀ y : M₂, (∀ x : M₁, B x y = 0) → y = 0 | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | SeparatingRight | A bilinear map is called right-separating if
the only element that is right-orthogonal to every other element is `0`; i.e.,
for every nonzero `y` in `M₂`, there exists `x` in `M₁` with `B x y ≠ 0`. |
Nondegenerate (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) : Prop :=
SeparatingLeft B ∧ SeparatingRight B
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | Nondegenerate | A bilinear map is called non-degenerate if it is left-separating and right-separating. |
flip_separatingRight {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.flip.SeparatingRight ↔ B.SeparatingLeft :=
⟨fun hB x hy ↦ hB x hy, fun hB x hy ↦ hB x hy⟩
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | flip_separatingRight | null |
flip_separatingLeft {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.flip.SeparatingLeft ↔ SeparatingRight B := by rw [← flip_separatingRight, flip_flip]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | flip_separatingLeft | null |
flip_nondegenerate {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} : B.flip.Nondegenerate ↔ B.Nondegenerate :=
Iff.trans and_comm (and_congr flip_separatingRight flip_separatingLeft) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | flip_nondegenerate | null |
separatingLeft_iff_linear_nontrivial {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.SeparatingLeft ↔ ∀ x : M₁, B x = 0 → x = 0 := by
constructor <;> intro h x hB
· simpa only [hB, zero_apply, eq_self_iff_true, forall_const] using h x
have h' : B x = 0 := by
ext
rw [zero_apply]
exact hB _
exact h x h' | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | separatingLeft_iff_linear_nontrivial | null |
separatingRight_iff_linear_flip_nontrivial {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.SeparatingRight ↔ ∀ y : M₂, B.flip y = 0 → y = 0 := by
rw [← flip_separatingLeft, separatingLeft_iff_linear_nontrivial] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | separatingRight_iff_linear_flip_nontrivial | null |
separatingLeft_iff_ker_eq_bot {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.SeparatingLeft ↔ LinearMap.ker B = ⊥ :=
Iff.trans separatingLeft_iff_linear_nontrivial LinearMap.ker_eq_bot'.symm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | separatingLeft_iff_ker_eq_bot | A bilinear map is left-separating if and only if it has a trivial kernel. |
separatingRight_iff_flip_ker_eq_bot {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.SeparatingRight ↔ LinearMap.ker B.flip = ⊥ := by
rw [← flip_separatingLeft, separatingLeft_iff_ker_eq_bot] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | separatingRight_iff_flip_ker_eq_bot | A bilinear map is right-separating if and only if its flip has a trivial kernel. |
IsRefl.nondegenerate_iff_separatingLeft {B : M →ₗ[R] M →ₗ[R] M₁} (hB : B.IsRefl) :
B.Nondegenerate ↔ B.SeparatingLeft := by
refine ⟨fun h ↦ h.1, fun hB' ↦ ⟨hB', ?_⟩⟩
rw [separatingRight_iff_flip_ker_eq_bot, hB.ker_eq_bot_iff_ker_flip_eq_bot.mp]
rwa [← separatingLeft_iff_ker_eq_bot] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsRefl.nondegenerate_iff_separatingLeft | null |
IsRefl.nondegenerate_iff_separatingRight {B : M →ₗ[R] M →ₗ[R] M₁} (hB : B.IsRefl) :
B.Nondegenerate ↔ B.SeparatingRight := by
refine ⟨fun h ↦ h.2, fun hB' ↦ ⟨?_, hB'⟩⟩
rw [separatingLeft_iff_ker_eq_bot, hB.ker_eq_bot_iff_ker_flip_eq_bot.mpr]
rwa [← separatingRight_iff_flip_ker_eq_bot] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsRefl.nondegenerate_iff_separatingRight | null |
disjoint_ker_of_nondegenerate_restrict {B : M →ₗ[R] M →ₗ[R] M₁} {W : Submodule R M}
(hW : (B.domRestrict₁₂ W W).Nondegenerate) :
Disjoint W (LinearMap.ker B) := by
refine Submodule.disjoint_def.mpr fun x hx hx' ↦ ?_
let x' : W := ⟨x, hx⟩
suffices x' = 0 by simpa [x']
apply hW.1 x'
simp_rw [Subtype.for... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | disjoint_ker_of_nondegenerate_restrict | null |
IsSymm.nondegenerate_restrict_of_isCompl_ker {B : M →ₗ[R] M →ₗ[R] R} (hB : B.IsSymm)
{W : Submodule R M} (hW : IsCompl W (LinearMap.ker B)) :
(B.domRestrict₁₂ W W).Nondegenerate := by
have hB' : (B.domRestrict₁₂ W W).IsRefl := fun x y ↦ hB.isRefl (W.subtype x) (W.subtype y)
rw [LinearMap.IsRefl.nondegenerat... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsSymm.nondegenerate_restrict_of_isCompl_ker | null |
nondegenerate_restrict_of_disjoint_orthogonal {B : M →ₗ[R] M →ₗ[R] M₁} (hB : B.IsRefl)
{W : Submodule R M} (hW : Disjoint W (W.orthogonalBilin B)) :
(B.domRestrict₁₂ W W).Nondegenerate := by
rw [(hB.domRestrict W).nondegenerate_iff_separatingLeft]
rintro ⟨x, hx⟩ b₁
rw [Submodule.mk_eq_zero, ← Submodule.me... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | nondegenerate_restrict_of_disjoint_orthogonal | The restriction of a reflexive bilinear map `B` onto a submodule `W` is
nondegenerate if `W` has trivial intersection with its orthogonal complement,
that is `Disjoint W (W.orthogonalBilin B)`. |
IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft [Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁} {v : Basis n R M} (h : B.IsOrthoᵢ v) (hB : B.SeparatingLeft)
(i : n) : ¬B.IsOrtho (v i) (v i) := by
intro ho
refine v.ne_zero i (hB (v i) fun m ↦ ?_)
obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m
rw [Basis.repr_s... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft | An orthogonal basis with respect to a left-separating bilinear map has no self-orthogonal
elements. |
IsOrthoᵢ.not_isOrtho_basis_self_of_separatingRight [Nontrivial R]
{B : M →ₛₗ[I] M →ₛₗ[I'] M₁} {v : Basis n R M} (h : B.IsOrthoᵢ v) (hB : B.SeparatingRight)
(i : n) : ¬B.IsOrtho (v i) (v i) := by
rw [isOrthoᵢ_flip] at h
rw [isOrtho_flip]
exact h.not_isOrtho_basis_self_of_separatingLeft (flip_separatingLeft... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthoᵢ.not_isOrtho_basis_self_of_separatingRight | An orthogonal basis with respect to a right-separating bilinear map has no self-orthogonal
elements. |
IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_self [NoZeroSMulDivisors R M₁]
{B : M →ₗ[R] M →ₗ[R] M₁} (v : Basis n R M) (hO : B.IsOrthoᵢ v)
(h : ∀ i, ¬B.IsOrtho (v i) (v i)) : B.SeparatingLeft := by
intro m hB
obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m
rw [LinearEquiv.map_eq_zero_iff]
ext i
rw [Finsu... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_self | Given an orthogonal basis with respect to a bilinear map, the bilinear map is left-separating if
the basis has no elements which are self-orthogonal. |
IsOrthoᵢ.separatingRight_iff_not_isOrtho_basis_self [NoZeroSMulDivisors R M₁]
{B : M →ₗ[R] M →ₗ[R] M₁} (v : Basis n R M) (hO : B.IsOrthoᵢ v)
(h : ∀ i, ¬B.IsOrtho (v i) (v i)) : B.SeparatingRight := by
rw [isOrthoᵢ_flip] at hO
rw [← flip_separatingLeft]
refine IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_s... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthoᵢ.separatingRight_iff_not_isOrtho_basis_self | Given an orthogonal basis with respect to a bilinear map, the bilinear map is right-separating
if the basis has no elements which are self-orthogonal. |
IsOrthoᵢ.nondegenerate_of_not_isOrtho_basis_self [NoZeroSMulDivisors R M₁]
{B : M →ₗ[R] M →ₗ[R] M₁} (v : Basis n R M) (hO : B.IsOrthoᵢ v)
(h : ∀ i, ¬B.IsOrtho (v i) (v i)) : B.Nondegenerate :=
⟨IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_self v hO h,
IsOrthoᵢ.separatingRight_iff_not_isOrtho_basis_self v ... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthoᵢ.nondegenerate_of_not_isOrtho_basis_self | Given an orthogonal basis with respect to a bilinear map, the bilinear map is nondegenerate
if the basis has no elements which are self-orthogonal. |
apply_smul_sub_smul_sub_eq [CommRing R] [AddCommGroup M] [Module R M]
(B : LinearMap.BilinForm R M) (x y : M) :
B ((B x y) • x - (B x x) • y) ((B x y) • x - (B x x) • y) =
(B x x) * ((B x x) * (B y y) - (B x y) * (B y x)) := by
simp only [map_sub, map_smul, sub_apply, smul_apply, smul_eq_mul, mul_sub,
... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | apply_smul_sub_smul_sub_eq | null |
apply_mul_apply_le_of_forall_zero_le (hs : ∀ x, 0 ≤ B x x) (x y : M) :
(B x y) * (B y x) ≤ (B x x) * (B y y) := by
have aux (x y : M) : 0 ≤ (B x x) * ((B x x) * (B y y) - (B x y) * (B y x)) := by
rw [← apply_smul_sub_smul_sub_eq B x y]
exact hs (B x y • x - B x x • y)
rcases lt_or_ge 0 (B x x) with hx |... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | apply_mul_apply_le_of_forall_zero_le | The **Cauchy-Schwarz inequality** for positive semidefinite forms. |
apply_sq_le_of_symm (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm) (x y : M) :
(B x y) ^ 2 ≤ (B x x) * (B y y) := by
rw [show (B x y) ^ 2 = (B x y) * (B y x) by rw [sq, ← hB.eq, RingHom.id_apply]]
exact apply_mul_apply_le_of_forall_zero_le B hs x y | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | apply_sq_le_of_symm | The **Cauchy-Schwarz inequality** for positive semidefinite symmetric forms. |
not_linearIndependent_of_apply_mul_apply_eq (hp : ∀ x, x ≠ 0 → 0 < B x x)
(x y : M) (he : (B x y) * (B y x) = (B x x) * (B y y)) :
¬ LinearIndependent R ![x, y] := by
have hz : (B x y) • x - (B x x) • y = 0 := by
by_contra hc
exact (ne_of_lt (hp ((B x) y • x - (B x) x • y) hc)).symm <|
(apply_sm... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | not_linearIndependent_of_apply_mul_apply_eq | The equality case of **Cauchy-Schwarz**. |
apply_mul_apply_lt_iff_linearIndependent [NoZeroSMulDivisors R M]
(hp : ∀ x, x ≠ 0 → 0 < B x x) (x y : M) :
(B x y) * (B y x) < (B x x) * (B y y) ↔ LinearIndependent R ![x, y] := by
have hle : ∀ z, 0 ≤ B z z := by
intro z
by_cases hz : z = 0; simp [hz]
exact le_of_lt (hp z hz)
constructor
· co... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | apply_mul_apply_lt_iff_linearIndependent | Strict **Cauchy-Schwarz** is equivalent to linear independence for positive definite forms. |
apply_sq_lt_iff_linearIndependent_of_symm [NoZeroSMulDivisors R M]
(hp : ∀ x, x ≠ 0 → 0 < B x x) (hB : B.IsSymm) (x y : M) :
(B x y) ^ 2 < (B x x) * (B y y) ↔ LinearIndependent R ![x, y] := by
rw [show (B x y) ^ 2 = (B x y) * (B y x) by rw [sq, ← hB.eq, RingHom.id_apply]]
exact apply_mul_apply_lt_iff_linear... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | apply_sq_lt_iff_linearIndependent_of_symm | Strict **Cauchy-Schwarz** is equivalent to linear independence for positive definite symmetric
forms. |
apply_apply_same_eq_zero_iff (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm) {x : M} :
B x x = 0 ↔ x ∈ LinearMap.ker B := by
rw [LinearMap.mem_ker]
refine ⟨fun h ↦ ?_, fun h ↦ by simp [h]⟩
ext y
have := B.apply_sq_le_of_symm hs hB x y
simp only [h, zero_mul] at this
exact pow_eq_zero <| le_antisymm this (sq_nonne... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | apply_apply_same_eq_zero_iff | null |
nondegenerate_iff (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm) :
B.Nondegenerate ↔ ∀ x, B x x = 0 ↔ x = 0 := by
simp_rw [hB.isRefl.nondegenerate_iff_separatingLeft, separatingLeft_iff_ker_eq_bot,
Submodule.eq_bot_iff, B.apply_apply_same_eq_zero_iff hs hB, mem_ker]
exact forall_congr' fun x ↦ by aesop | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | nondegenerate_iff | null |
nondegenerate_iff' (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm) :
B.Nondegenerate ↔ ∀ x, x ≠ 0 → 0 < B x x := by
rw [B.nondegenerate_iff hs hB, ← not_iff_not]
push_neg
exact exists_congr fun x ↦ ⟨by aesop, fun ⟨h₀, h⟩ ↦ Or.inl ⟨le_antisymm h (hs x), h₀⟩⟩ | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | nondegenerate_iff' | A convenience variant of `LinearMap.BilinForm.nondegenerate_iff` characterising nondegeneracy as
positive definiteness. |
nondegenerate_restrict_iff_disjoint_ker (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm)
{W : Submodule R M} :
(B.domRestrict₁₂ W W).Nondegenerate ↔ Disjoint W (LinearMap.ker B) := by
refine ⟨disjoint_ker_of_nondegenerate_restrict, fun hW ↦ ?_⟩
have hB' : (B.domRestrict₁₂ W W).IsRefl := fun x y ↦ hB.isRefl (W.subtype ... | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | nondegenerate_restrict_iff_disjoint_ker | null |
SModEq (x y : M) : Prop :=
(Submodule.Quotient.mk x : M ⧸ U) = Submodule.Quotient.mk y
@[inherit_doc] notation:50 x " ≡ " y " [SMOD " N "]" => SModEq N x y
variable {U U₁ U₂} | def | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | SModEq | A predicate saying two elements of a module are equivalent modulo a submodule. |
protected SModEq.def :
x ≡ y [SMOD U] ↔ (Submodule.Quotient.mk x : M ⧸ U) = Submodule.Quotient.mk y :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | SModEq.def | null |
sub_mem : x ≡ y [SMOD U] ↔ x - y ∈ U := by rw [SModEq.def, Submodule.Quotient.eq]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | sub_mem | null |
top : x ≡ y [SMOD (⊤ : Submodule R M)] :=
(Submodule.Quotient.eq ⊤).2 mem_top
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | top | null |
bot : x ≡ y [SMOD (⊥ : Submodule R M)] ↔ x = y := by
rw [SModEq.def, Submodule.Quotient.eq, mem_bot, sub_eq_zero]
@[mono] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | bot | null |
mono (HU : U₁ ≤ U₂) (hxy : x ≡ y [SMOD U₁]) : x ≡ y [SMOD U₂] :=
(Submodule.Quotient.eq U₂).2 <| HU <| (Submodule.Quotient.eq U₁).1 hxy
@[refl] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | mono | null |
protected refl (x : M) : x ≡ x [SMOD U] :=
@rfl _ _ | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | refl | null |
protected rfl : x ≡ x [SMOD U] :=
SModEq.refl _ | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | rfl | null |
comm : x ≡ y [SMOD U] ↔ y ≡ x [SMOD U] := ⟨symm, symm⟩
@[trans]
nonrec theorem trans (hxy : x ≡ y [SMOD U]) (hyz : y ≡ z [SMOD U]) : x ≡ z [SMOD U] :=
hxy.trans hyz | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | comm | null |
instTrans : Trans (SModEq U) (SModEq U) (SModEq U) where
trans := trans
@[gcongr] | instance | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | instTrans | null |
add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by
rw [SModEq.def] at hxy₁ hxy₂ ⊢
simp_rw [Quotient.mk_add, hxy₁, hxy₂]
@[gcongr] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | add | null |
sum {ι} {s : Finset ι} {x y : ι → M}
(hxy : ∀ i ∈ s, x i ≡ y i [SMOD U]) : ∑ i ∈ s, x i ≡ ∑ i ∈ s, y i [SMOD U] := by
classical
induction s using Finset.cons_induction with
| empty => simp [SModEq.rfl]
| cons i s _ ih =>
grw [Finset.sum_cons, Finset.sum_cons, hxy i (Finset.mem_cons_self i s),
ih (... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | sum | null |
smul (hxy : x ≡ y [SMOD U]) (c : R) : c • x ≡ c • y [SMOD U] := by
rw [SModEq.def] at hxy ⊢
simp_rw [Quotient.mk_smul, hxy]
@[gcongr] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | smul | null |
nsmul (hxy : x ≡ y [SMOD U]) (n : ℕ) : n • x ≡ n • y [SMOD U] := by
rw [SModEq.def] at hxy ⊢
simp_rw [Quotient.mk_smul, hxy]
@[gcongr] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | nsmul | null |
zsmul (hxy : x ≡ y [SMOD U]) (n : ℤ) : n • x ≡ n • y [SMOD U] := by
rw [SModEq.def] at hxy ⊢
simp_rw [Quotient.mk_smul, hxy]
@[gcongr] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | zsmul | null |
mul {I : Ideal A} {x₁ x₂ y₁ y₂ : A} (hxy₁ : x₁ ≡ y₁ [SMOD I])
(hxy₂ : x₂ ≡ y₂ [SMOD I]) : x₁ * x₂ ≡ y₁ * y₂ [SMOD I] := by
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_mul] at hxy₁ hxy₂ ⊢
rw [hxy₁, hxy₂]
@[gcongr] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | mul | null |
prod {I : Ideal A} {ι} {s : Finset ι} {x y : ι → A}
(hxy : ∀ i ∈ s, x i ≡ y i [SMOD I]) : ∏ i ∈ s, x i ≡ ∏ i ∈ s, y i [SMOD I] := by
classical
induction s using Finset.cons_induction with
| empty => simp [SModEq.rfl]
| cons i s _ ih =>
grw [Finset.prod_cons, Finset.prod_cons, hxy i (Finset.mem_cons_self... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | prod | null |
pow {I : Ideal A} {x y : A} (n : ℕ) (hxy : x ≡ y [SMOD I]) :
x ^ n ≡ y ^ n [SMOD I] := by
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy]
@[gcongr] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | pow | null |
neg (hxy : x ≡ y [SMOD U]) : - x ≡ - y [SMOD U] := by
simpa only [SModEq.def, Quotient.mk_neg, neg_inj]
@[gcongr] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | neg | null |
sub (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ - x₂ ≡ y₁ - y₂ [SMOD U] := by
rw [SModEq.def] at hxy₁ hxy₂ ⊢
simp_rw [Quotient.mk_sub, hxy₁, hxy₂] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | sub | null |
zero : x ≡ 0 [SMOD U] ↔ x ∈ U := by rw [SModEq.def, Submodule.Quotient.eq, sub_zero] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | zero | null |
_root_.sub_smodEq_zero : x - y ≡ 0 [SMOD U] ↔ x ≡ y [SMOD U] := by
simp only [SModEq.sub_mem, sub_zero] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | _root_.sub_smodEq_zero | null |
map (hxy : x ≡ y [SMOD U]) (f : M →ₗ[R] N) : f x ≡ f y [SMOD U.map f] :=
(Submodule.Quotient.eq _).2 <| f.map_sub x y ▸ mem_map_of_mem <| (Submodule.Quotient.eq _).1 hxy | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | map | null |
comap {f : M →ₗ[R] N} (hxy : f x ≡ f y [SMOD V]) : x ≡ y [SMOD V.comap f] :=
(Submodule.Quotient.eq _).2 <|
show f (x - y) ∈ V from (f.map_sub x y).symm ▸ (Submodule.Quotient.eq _).1 hxy
@[gcongr] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | comap | null |
eval {R : Type*} [CommRing R] {I : Ideal R} {x y : R} (h : x ≡ y [SMOD I]) (f : R[X]) :
f.eval x ≡ f.eval y [SMOD I] := by
simp_rw [Polynomial.eval_eq_sum, Polynomial.sum]
gcongr | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] | Mathlib/LinearAlgebra/SModEq.lean | eval | null |
linearIndependent_single [Semiring R] [∀ i, AddCommMonoid (Ms i)] [∀ i, Module R (Ms i)]
[DecidableEq η] (v : ∀ j, ιs j → Ms j) (hs : ∀ i, LinearIndependent R (v i)) :
LinearIndependent R fun ji : Σ j, ιs j ↦ Pi.single ji.1 (v ji.1 ji.2) := by
convert (DFinsupp.linearIndependent_single _ hs).map_injOn _ DFins... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | linearIndependent_single | null |
linearIndependent_single_one (ι R : Type*) [Semiring R] [DecidableEq ι] :
LinearIndependent R (fun i : ι ↦ Pi.single i (1 : R)) := by
rw [← linearIndependent_equiv (Equiv.sigmaPUnit ι)]
exact Pi.linearIndependent_single (fun (_ : ι) (_ : Unit) ↦ (1 : R))
<| by simp +contextual [Fintype.linearIndependent_iff... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | linearIndependent_single_one | null |
linearIndependent_single_of_ne_zero {ι R M : Type*} [Ring R] [AddCommGroup M] [Module R M]
[NoZeroSMulDivisors R M] [DecidableEq ι] {v : ι → M} (hv : ∀ i, v i ≠ 0) :
LinearIndependent R fun i : ι ↦ Pi.single i (v i) := by
rw [← linearIndependent_equiv (Equiv.sigmaPUnit ι)]
exact linearIndependent_single (fu... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | linearIndependent_single_of_ne_zero | null |
linearIndependent_single_ne_zero {ι R : Type*} [Ring R] [NoZeroDivisors R] [DecidableEq ι]
{v : ι → R} (hv : ∀ i, v i ≠ 0) : LinearIndependent R (fun i : ι ↦ Pi.single i (v i)) :=
linearIndependent_single_of_ne_zero hv
variable [Semiring R] [∀ i, AddCommMonoid (Ms i)] [∀ i, Module R (Ms i)] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | linearIndependent_single_ne_zero | null |
protected noncomputable basis (s : ∀ j, Basis (ιs j) R (Ms j)) :
Basis (Σ j, ιs j) R (∀ j, Ms j) :=
Basis.ofRepr
((LinearEquiv.piCongrRight fun j => (s j).repr) ≪≫ₗ
(Finsupp.sigmaFinsuppLEquivPiFinsupp R).symm)
@[simp] | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basis | `Pi.basis (s : ∀ j, Basis (ιs j) R (Ms j))` is the `Σ j, ιs j`-indexed basis on `Π j, Ms j`
given by `s j` on each component.
For the standard basis over `R` on the finite-dimensional space `η → R` see `Pi.basisFun`. |
basis_repr_single [DecidableEq η] (s : ∀ j, Basis (ιs j) R (Ms j)) (j i) :
(Pi.basis s).repr (Pi.single j (s j i)) = Finsupp.single ⟨j, i⟩ 1 := by
classical
ext ⟨j', i'⟩
by_cases hj : j = j'
· subst hj
simp only [Pi.basis, LinearEquiv.trans_apply,
LinearEquiv.piCongrRight, Finsupp.sigmaFinsuppLEqu... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basis_repr_single | null |
basis_apply [DecidableEq η] (s : ∀ j, Basis (ιs j) R (Ms j)) (ji) :
Pi.basis s ji = Pi.single ji.1 (s ji.1 ji.2) :=
Basis.apply_eq_iff.mpr (by simp)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basis_apply | null |
basis_repr (s : ∀ j, Basis (ιs j) R (Ms j)) (x) (ji) :
(Pi.basis s).repr x ji = (s ji.1).repr (x ji.1) ji.2 :=
rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basis_repr | null |
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