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 ⌀ |
|---|---|---|---|---|---|---|
exists_nonneg_right (h : SameRay R x y) (hy : y ≠ 0) : ∃ r : R, 0 ≤ r ∧ x = r • y :=
(h.symm.exists_nonneg_left hy).imp fun _ => And.imp_right Eq.symm | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_nonneg_right | If a vector `v₁` is on the same ray as a nonzero vector `v₂`, then it is equal to `c • v₂` for
some nonnegative `c`. |
exists_eq_smul_add (h : SameRay R v₁ v₂) :
∃ a b : R, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • (v₁ + v₂) ∧ v₂ = b • (v₁ + v₂) := by
rcases h with (rfl | rfl | ⟨r₁, r₂, h₁, h₂, H⟩)
· use 0, 1
simp
· use 1, 0
simp
· have h₁₂ : 0 < r₁ + r₂ := add_pos h₁ h₂
refine
⟨r₂ / (r₁ + r₂), r₁ / (r₁ + r₂), ... | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_eq_smul_add | If vectors `v₁` and `v₂` are on the same ray, then for some nonnegative `a b`, `a + b = 1`, we
have `v₁ = a • (v₁ + v₂)` and `v₂ = b • (v₁ + v₂)`. |
exists_eq_smul (h : SameRay R v₁ v₂) :
∃ (u : M) (a b : R), 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ v₁ = a • u ∧ v₂ = b • u :=
⟨v₁ + v₂, h.exists_eq_smul_add⟩ | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_eq_smul | If vectors `v₁` and `v₂` are on the same ray, then they are nonnegative multiples of the same
vector. Actually, this vector can be assumed to be `v₁ + v₂`, see `SameRay.exists_eq_smul_add`. |
exists_pos_left_iff_sameRay (hx : x ≠ 0) (hy : y ≠ 0) :
(∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y := by
refine ⟨fun h => ?_, fun h => h.exists_pos_left hx hy⟩
rcases h with ⟨r, hr, rfl⟩
exact SameRay.sameRay_pos_smul_right x hr | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_pos_left_iff_sameRay | null |
exists_pos_left_iff_sameRay_and_ne_zero (hx : x ≠ 0) :
(∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y ∧ y ≠ 0 := by
constructor
· rintro ⟨r, hr, rfl⟩
simp [hx, hr.le, hr.ne']
· rintro ⟨hxy, hy⟩
exact (exists_pos_left_iff_sameRay hx hy).2 hxy | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_pos_left_iff_sameRay_and_ne_zero | null |
exists_nonneg_left_iff_sameRay (hx : x ≠ 0) :
(∃ r : R, 0 ≤ r ∧ r • x = y) ↔ SameRay R x y := by
refine ⟨fun h => ?_, fun h => h.exists_nonneg_left hx⟩
rcases h with ⟨r, hr, rfl⟩
exact SameRay.sameRay_nonneg_smul_right x hr | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_nonneg_left_iff_sameRay | null |
exists_pos_right_iff_sameRay (hx : x ≠ 0) (hy : y ≠ 0) :
(∃ r : R, 0 < r ∧ x = r • y) ↔ SameRay R x y := by
rw [SameRay.sameRay_comm]
simp_rw [eq_comm (a := x)]
exact exists_pos_left_iff_sameRay hy hx | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_pos_right_iff_sameRay | null |
exists_pos_right_iff_sameRay_and_ne_zero (hy : y ≠ 0) :
(∃ r : R, 0 < r ∧ x = r • y) ↔ SameRay R x y ∧ x ≠ 0 := by
rw [SameRay.sameRay_comm]
simp_rw [eq_comm (a := x)]
exact exists_pos_left_iff_sameRay_and_ne_zero hy | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_pos_right_iff_sameRay_and_ne_zero | null |
exists_nonneg_right_iff_sameRay (hy : y ≠ 0) :
(∃ r : R, 0 ≤ r ∧ x = r • y) ↔ SameRay R x y := by
rw [SameRay.sameRay_comm]
simp_rw [eq_comm (a := x)]
exact exists_nonneg_left_iff_sameRay (R := R) hy | theorem | LinearAlgebra | [
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Algebra.Order.Module.Algebra",
"Mathlib.Algebra.Ring.Subring.Units",
"Mathlib.LinearAlgebra.LinearIndependent.Defs",
"Mathlib.Tactic.LinearCombination",
"Mathlib.Tactic.Module",
"Mathlib.Tactic.Positivity.Basic"
] | Mathlib/LinearAlgebra/Ray.lean | exists_nonneg_right_iff_sameRay | null |
preReflection : End R M :=
LinearMap.id - f.smulRight x | def | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | preReflection | Given an element `x` in a module `M` and a linear form `f` on `M`, we define the endomorphism
of `M` for which `y ↦ y - (f y) • x`.
One is typically interested in this endomorphism when `f x = 2`; this definition exists to allow the
user defer discharging this proof obligation. See also `Module.reflection`. |
preReflection_apply :
preReflection x f y = y - (f y) • x := by
simp [preReflection]
variable {x f} | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | preReflection_apply | null |
preReflection_apply_self (h : f x = 2) :
preReflection x f x = - x := by
rw [preReflection_apply, h, two_smul]; abel | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | preReflection_apply_self | null |
involutive_preReflection (h : f x = 2) :
Involutive (preReflection x f) :=
fun y ↦ by simp [map_sub, h, two_smul, preReflection_apply] | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | involutive_preReflection | null |
preReflection_preReflection (g : Dual R M) (h : f x = 2) :
preReflection (preReflection x f y) (preReflection f (Dual.eval R M x) g) =
(preReflection x f) ∘ₗ (preReflection y g) ∘ₗ (preReflection x f) := by
ext m
simp only [h, preReflection_apply, mul_comm (g x) (f m), mul_two, mul_assoc, Dual.eval_apply,
... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | preReflection_preReflection | null |
reflection (h : f x = 2) : M ≃ₗ[R] M :=
{ preReflection x f, (involutive_preReflection h).toPerm with } | def | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection | Given an element `x` in a module `M` and a linear form `f` on `M` for which `f x = 2`, we define
the endomorphism of `M` for which `y ↦ y - (f y) • x`.
It is an involutive endomorphism of `M` fixing the kernel of `f` for which `x ↦ -x`. |
reflection_apply (h : f x = 2) :
reflection h y = y - (f y) • x :=
preReflection_apply x f y
@[simp] | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_apply | null |
reflection_apply_self (h : f x = 2) :
reflection h x = - x :=
preReflection_apply_self h | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_apply_self | null |
involutive_reflection (h : f x = 2) :
Involutive (reflection h) :=
involutive_preReflection h
@[simp] | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | involutive_reflection | null |
reflection_inv (h : f x = 2) : (reflection h)⁻¹ = reflection h := rfl
@[simp] | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_inv | null |
reflection_symm (h : f x = 2) :
(reflection h).symm = reflection h :=
rfl | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_symm | null |
invOn_reflection_of_mapsTo {Φ : Set M} (h : f x = 2) :
InvOn (reflection h) (reflection h) Φ Φ :=
⟨fun x _ ↦ involutive_reflection h x, fun x _ ↦ involutive_reflection h x⟩ | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | invOn_reflection_of_mapsTo | null |
bijOn_reflection_of_mapsTo {Φ : Set M} (h : f x = 2) (h' : MapsTo (reflection h) Φ Φ) :
BijOn (reflection h) Φ Φ :=
(invOn_reflection_of_mapsTo h).bijOn h' h' | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | bijOn_reflection_of_mapsTo | null |
_root_.Submodule.mem_invtSubmodule_reflection_of_mem (h : f x = 2)
(p : Submodule R M) (hx : x ∈ p) :
p ∈ End.invtSubmodule (reflection h) := by
suffices ∀ y ∈ p, reflection h y ∈ p from
(End.mem_invtSubmodule _).mpr fun y hy ↦ by simpa using this y hy
intro y hy
simpa only [reflection_apply, p.sub_me... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | _root_.Submodule.mem_invtSubmodule_reflection_of_mem | null |
_root_.Submodule.mem_invtSubmodule_reflection_iff [NeZero (2 : R)] [NoZeroSMulDivisors R M]
(h : f x = 2) {p : Submodule R M} (hp : Disjoint p (R ∙ x)) :
p ∈ End.invtSubmodule (reflection h) ↔ p ≤ LinearMap.ker f := by
refine ⟨fun h' y hy ↦ ?_, fun h' y hy ↦ ?_⟩
· have hx : x ≠ 0 := by rintro rfl; exact two... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | _root_.Submodule.mem_invtSubmodule_reflection_iff | null |
reflection_mul_reflection_pow_apply (m : ℕ) (z : M)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
((reflection hf * reflection hg) ^ m) z =
z +
((S R ((m - 2) / 2)).eval t * ((S R ((m - 1) / 2)).eval t + (S R ((m - 3) / 2)).eval t)) •
((g x * f z - g z) • y - f z • x) +
... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_pow_apply | A formula for $(r_1 r_2)^m z$, where $m$ is a natural number and $z \in M$. |
reflection_mul_reflection_pow (m : ℕ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
((reflection hf * reflection hg) ^ m).toLinearMap =
LinearMap.id (R := R) (M := M) +
((S R ((m - 2) / 2)).eval t * ((S R ((m - 1) / 2)).eval t + (S R ((m - 3) / 2)).eval t)) •
((g x • f - ... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_pow | A formula for $(r_1 r_2)^m$, where $m$ is a natural number. |
reflection_mul_reflection_zpow_apply (m : ℤ) (z : M)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
((reflection hf * reflection hg) ^ m) z =
z +
((S R ((m - 2) / 2)).eval t * ((S R ((m - 1) / 2)).eval t + (S R ((m - 3) / 2)).eval t)) •
((g x * f z - g z) • y - f z • x) +
... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_zpow_apply | A formula for $(r_1 r_2)^m z$, where $m$ is an integer and $z \in M$. |
reflection_mul_reflection_zpow (m : ℤ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
((reflection hf * reflection hg) ^ m).toLinearMap =
LinearMap.id (R := R) (M := M) +
((S R ((m - 2) / 2)).eval t * ((S R ((m - 1) / 2)).eval t + (S R ((m - 3) / 2)).eval t)) •
((g x • f -... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_zpow | A formula for $(r_1 r_2)^m$, where $m$ is an integer. |
reflection_mul_reflection_zpow_apply_self (m : ℤ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
((reflection hf * reflection hg) ^ m) x =
((S R m).eval t + (S R (m - 1)).eval t) • x + ((S R (m - 1)).eval t * -g x) • y := by
/- Even though this is a special case of `Module.reflection_mul_... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_zpow_apply_self | A formula for $(r_1 r_2)^m x$, where $m$ is an integer. This is the special case of
`Module.reflection_mul_reflection_zpow_apply` with $z = x$. |
reflection_mul_reflection_pow_apply_self (m : ℕ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
((reflection hf * reflection hg) ^ m) x =
((S R m).eval t + (S R (m - 1)).eval t) • x + ((S R (m - 1)).eval t * -g x) • y :=
mod_cast reflection_mul_reflection_zpow_apply_self hf hg m t ht | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_pow_apply_self | A formula for $(r_1 r_2)^m x$, where $m$ is a natural number. This is the special case of
`Module.reflection_mul_reflection_pow_apply` with $z = x$. |
reflection_mul_reflection_mul_reflection_zpow_apply_self (m : ℤ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
(reflection hg * (reflection hf * reflection hg) ^ m) x =
((S R m).eval t + (S R (m - 1)).eval t) • x + ((S R m).eval t * -g x) • y := by
rw [LinearEquiv.mul_apply, reflection_m... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_mul_reflection_zpow_apply_self | A formula for $r_2 (r_1 r_2)^m x$, where $m$ is an integer. |
reflection_mul_reflection_mul_reflection_pow_apply_self (m : ℕ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 := by rfl) :
(reflection hg * (reflection hf * reflection hg) ^ m) x =
((S R m).eval t + (S R (m - 1)).eval t) • x + ((S R m).eval t * -g x) • y :=
mod_cast reflection_mul_reflection_mul_refl... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_mul_reflection_mul_reflection_pow_apply_self | A formula for $r_2 (r_1 r_2)^m x$, where $m$ is a natural number. |
Dual.eq_of_preReflection_mapsTo [CharZero R] [NoZeroSMulDivisors R M]
{x : M} {Φ : Set M} (hΦ₁ : Φ.Finite) (hΦ₂ : span R Φ = ⊤) {f g : Dual R M}
(hf₁ : f x = 2) (hf₂ : MapsTo (preReflection x f) Φ Φ)
(hg₁ : g x = 2) (hg₂ : MapsTo (preReflection x g) Φ Φ) :
f = g := by
have hx : x ≠ 0 := by rintro rfl;... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | Dual.eq_of_preReflection_mapsTo | See also `Module.Dual.eq_of_preReflection_mapsTo'` for a variant of this lemma which
applies when `Φ` does not span.
This rather technical-looking lemma exists because it is exactly what is needed to establish various
uniqueness results for root data / systems. One might regard this lemma as lying at the boundary of
l... |
Dual.eq_of_preReflection_mapsTo' [CharZero R] [NoZeroSMulDivisors R M]
{x : M} {Φ : Set M} (hΦ₁ : Φ.Finite) (hx : x ∈ span R Φ) {f g : Dual R M}
(hf₁ : f x = 2) (hf₂ : MapsTo (preReflection x f) Φ Φ)
(hg₁ : g x = 2) (hg₂ : MapsTo (preReflection x g) Φ Φ) :
(span R Φ).subtype.dualMap f = (span R Φ).subty... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | Dual.eq_of_preReflection_mapsTo' | This rather technical-looking lemma exists because it is exactly what is needed to establish a
uniqueness result for root data. See the doc string of `Module.Dual.eq_of_preReflection_mapsTo` for
further remarks. |
reflection_reflection_iterate
(hfx : f x = 2) (hgy : g y = 2) (hgxfy : f y * g x = 4) (n : ℕ) :
((reflection hgy).trans (reflection hfx))^[n] y = y + n • (f y • x - (2 : R) • y) := by
induction n with
| zero => simp
| succ n ih =>
have hz : ∀ z : M, f y • g x • z = 2 • 2 • z := by
intro z
... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | reflection_reflection_iterate | Composite of reflections in "parallel" hyperplanes is a shear (special case). |
infinite_range_reflection_reflection_iterate_iff [NoZeroSMulDivisors ℤ M]
(hfx : f x = 2) (hgy : g y = 2) (hgxfy : f y * g x = 4) :
(range <| fun n ↦ ((reflection hgy).trans (reflection hfx))^[n] y).Infinite ↔
f y • x ≠ (2 : R) • y := by
simp only [reflection_reflection_iterate hfx hgy hgxfy, infinite_ran... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | infinite_range_reflection_reflection_iterate_iff | null |
eq_of_mapsTo_reflection_of_mem [NoZeroSMulDivisors ℤ M] {Φ : Set M} (hΦ : Φ.Finite)
(hfx : f x = 2) (hgy : g y = 2) (hgx : g x = 2) (hfy : f y = 2)
(hxfΦ : MapsTo (preReflection x f) Φ Φ)
(hygΦ : MapsTo (preReflection y g) Φ Φ)
(hyΦ : y ∈ Φ) :
x = y := by
suffices h : f y • x = (2 : R) • y by
... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | eq_of_mapsTo_reflection_of_mem | null |
injOn_dualMap_subtype_span_range_range {ι : Type*} [NoZeroSMulDivisors ℤ M]
{r : ι ↪ M} {c : ι → Dual R M} (hfin : (range r).Finite)
(h_two : ∀ i, c i (r i) = 2)
(h_mapsTo : ∀ i, MapsTo (preReflection (r i) (c i)) (range r) (range r)) :
InjOn (span R (range r)).subtype.dualMap (range c) := by
rintro -... | lemma | LinearAlgebra | [
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Algebra.EuclideanDomain.Int",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.Submodule.Invariant",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.LinearAlgebra.Dual.Defs",
"Mathlib.LinearAlgebra.Fini... | Mathlib/LinearAlgebra/Reflection.lean | injOn_dualMap_subtype_span_range_range | null |
IsSemisimple := IsSemisimpleModule R[X] (AEval' f) | def | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple | A linear endomorphism of an `R`-module `M` is called *semisimple* if the induced `R[X]`-module
structure on `M` is semisimple. This is equivalent to saying that every `f`-invariant `R`-submodule
of `M` has an `f`-invariant complement: see `Module.End.isSemisimple_iff`. |
IsFinitelySemisimple : Prop :=
∀ p (hp : p ∈ invtSubmodule f), Module.Finite R p → IsSemisimple (LinearMap.restrict f hp)
variable {f} | def | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsFinitelySemisimple | A weaker version of semisimplicity that only prescribes behaviour on finitely-generated
submodules. |
isSemisimple_iff' :
f.IsSemisimple ↔ ∀ p : invtSubmodule f, ∃ q : invtSubmodule f, IsCompl p q := by
rw [IsSemisimple, isSemisimpleModule_iff, (AEval.mapSubmodule R M f).symm.complementedLattice_iff,
complementedLattice_iff]
rfl | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_iff' | A linear endomorphism is semisimple if every invariant submodule has in invariant complement.
See also `Module.End.isSemisimple_iff`. |
isSemisimple_iff :
f.IsSemisimple ↔ ∀ p ∈ invtSubmodule f, ∃ q ∈ invtSubmodule f, IsCompl p q := by
simp [isSemisimple_iff'] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_iff | null |
isSemisimple_restrict_iff (p) (hp : p ∈ invtSubmodule f) :
IsSemisimple (LinearMap.restrict f hp) ↔
∀ q ∈ f.invtSubmodule, q ≤ p → ∃ r ≤ p, r ∈ f.invtSubmodule ∧ Disjoint q r ∧ q ⊔ r = p := by
let e : Submodule R[X] (AEval' (f.restrict hp)) ≃o Iic (AEval.mapSubmodule R M f ⟨p, hp⟩) :=
(Submodule.orderIsoM... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_restrict_iff | null |
isFinitelySemisimple_iff' :
f.IsFinitelySemisimple ↔ ∀ p (hp : p ∈ invtSubmodule f),
Module.Finite R p → IsSemisimple (LinearMap.restrict f hp) :=
Iff.rfl | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isFinitelySemisimple_iff' | A linear endomorphism is finitely semisimple if it is semisimple on every finitely-generated
invariant submodule.
See also `Module.End.isFinitelySemisimple_iff`. |
isFinitelySemisimple_iff :
f.IsFinitelySemisimple ↔ ∀ p ∈ invtSubmodule f, Module.Finite R p → ∀ q ∈ invtSubmodule f,
q ≤ p → ∃ r, r ≤ p ∧ r ∈ invtSubmodule f ∧ Disjoint q r ∧ q ⊔ r = p := by
simp_rw [isFinitelySemisimple_iff', isSemisimple_restrict_iff]
@[simp] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isFinitelySemisimple_iff | A characterisation of `Module.End.IsFinitelySemisimple` using only the lattice of submodules of
`M` (thus avoiding submodules of submodules). |
isSemisimple_zero [IsSemisimpleModule R M] : IsSemisimple (0 : Module.End R M) := by
simpa [isSemisimple_iff] using exists_isCompl
@[simp] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_zero | null |
isSemisimple_id [IsSemisimpleModule R M] : IsSemisimple (LinearMap.id : Module.End R M) := by
simpa [isSemisimple_iff] using exists_isCompl
@[simp] lemma isSemisimple_neg : (-f).IsSemisimple ↔ f.IsSemisimple := by
simp [isSemisimple_iff, mem_invtSubmodule]
variable (f) in | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_id | null |
protected _root_.LinearEquiv.isSemisimple_iff {M₂ : Type*} [AddCommGroup M₂] [Module R M₂]
(g : End R M₂) (e : M ≃ₗ[R] M₂) (he : e ∘ₗ f = g ∘ₗ e) :
f.IsSemisimple ↔ g.IsSemisimple := by
let e : AEval' f ≃ₗ[R[X]] AEval' g := LinearEquiv.ofAEval _ (e.trans (AEval'.of g)) fun x ↦ by
simpa [AEval'.X_smul_of] ... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | _root_.LinearEquiv.isSemisimple_iff | null |
eq_zero_of_isNilpotent_isSemisimple (hn : IsNilpotent f) (hs : f.IsSemisimple) : f = 0 := by
have ⟨n, h0⟩ := hn
rw [← aeval_X (R := R) f]; rw [← aeval_X_pow (R := R) f] at h0
rw [← RingHom.mem_ker, ← AEval.annihilator_eq_ker_aeval (M := M)] at h0 ⊢
exact hs.annihilator_isRadical _ _ ⟨n, h0⟩ | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | eq_zero_of_isNilpotent_isSemisimple | null |
eq_zero_of_isNilpotent_of_isFinitelySemisimple
(hn : IsNilpotent f) (hs : IsFinitelySemisimple f) : f = 0 := by
have (p) (hp₁ : p ∈ f.invtSubmodule) (hp₂ : Module.Finite R p) : f.restrict hp₁ = 0 := by
specialize hs p hp₁ hp₂
replace hn : IsNilpotent (f.restrict hp₁) := isNilpotent.restrict hp₁ hn
exa... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | eq_zero_of_isNilpotent_of_isFinitelySemisimple | null |
isSemisimple_sub_algebraMap_iff {μ : R} :
(f - algebraMap R (End R M) μ).IsSemisimple ↔ f.IsSemisimple := by
suffices ∀ p : Submodule R M, p ≤ p.comap (f - algebraMap R (Module.End R M) μ) ↔ p ≤ p.comap f by
simp [mem_invtSubmodule, isSemisimple_iff, this]
refine fun p ↦ ⟨fun h x hx ↦ ?_, fun h x hx ↦ p.sub... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_sub_algebraMap_iff | null |
IsSemisimple.restrict {p : Submodule R M} (hp : p ∈ f.invtSubmodule) (hf : f.IsSemisimple) :
IsSemisimple (f.restrict hp) := by
rw [IsSemisimple] at hf ⊢
let e : Submodule R[X] (AEval' (LinearMap.restrict f hp)) ≃o
Iic (AEval.mapSubmodule R M f ⟨p, hp⟩) :=
(Submodule.orderIsoMapComap <| AEval.restrict... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.restrict | null |
IsSemisimple.isFinitelySemisimple (hf : f.IsSemisimple) :
f.IsFinitelySemisimple :=
isFinitelySemisimple_iff'.mp fun _ _ _ ↦ hf.restrict _
@[simp] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.isFinitelySemisimple | null |
isFinitelySemisimple_iff_isSemisimple [Module.Finite R M] :
f.IsFinitelySemisimple ↔ f.IsSemisimple := by
refine ⟨fun hf ↦ isSemisimple_iff.mpr fun p hp ↦ ?_, IsSemisimple.isFinitelySemisimple⟩
obtain ⟨q, -, hq₁, hq₂, hq₃⟩ :=
isFinitelySemisimple_iff.mp hf ⊤ (invtSubmodule.top_mem f) inferInstance p hp le_t... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isFinitelySemisimple_iff_isSemisimple | null |
isFinitelySemisimple_sub_algebraMap_iff {μ : R} :
(f - algebraMap R (End R M) μ).IsFinitelySemisimple ↔ f.IsFinitelySemisimple := by
suffices ∀ p : Submodule R M, p ≤ p.comap (f - algebraMap R (Module.End R M) μ) ↔ p ≤ p.comap f by
simp_rw [isFinitelySemisimple_iff, mem_invtSubmodule, this]
refine fun p ↦ ⟨... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isFinitelySemisimple_sub_algebraMap_iff | null |
IsFinitelySemisimple.restrict {p : Submodule R M} (hp : p ∈ f.invtSubmodule)
(hf : f.IsFinitelySemisimple) :
IsFinitelySemisimple (f.restrict hp) := by
intro q hq₁ hq₂
have := invtSubmodule.map_subtype_mem_of_mem_invtSubmodule f hp hq₁
let e : q ≃ₗ[R] q.map p.subtype := p.equivSubtypeMap q
rw [e.isSemis... | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsFinitelySemisimple.restrict | null |
IsSemisimple_smul_iff {t : K} (ht : t ≠ 0) :
(t • f).IsSemisimple ↔ f.IsSemisimple := by
simp [isSemisimple_iff, mem_invtSubmodule, Submodule.comap_smul f (h := ht)] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple_smul_iff | null |
IsSemisimple_smul (t : K) (h : f.IsSemisimple) :
(t • f).IsSemisimple := by
wlog ht : t ≠ 0; · simp [not_not.mp ht]
rwa [IsSemisimple_smul_iff ht] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple_smul | null |
isSemisimple_of_squarefree_aeval_eq_zero {p : K[X]}
(hp : Squarefree p) (hpf : aeval f p = 0) : f.IsSemisimple := by
rw [← RingHom.mem_ker, ← AEval.annihilator_eq_ker_aeval (M := M), mem_annihilator,
← IsTorsionBy, ← isTorsionBySet_singleton_iff, isTorsionBySet_iff_is_torsion_by_span] at hpf
let R := K[X]... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | isSemisimple_of_squarefree_aeval_eq_zero | null |
IsSemisimple.minpoly_squarefree : Squarefree (minpoly K f) :=
IsRadical.squarefree (minpoly.ne_zero <| Algebra.IsIntegral.isIntegral _) <| by
rw [isRadical_iff_span_singleton, span_minpoly_eq_annihilator]; exact hf.annihilator_isRadical | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.minpoly_squarefree | The minimal polynomial of a semisimple endomorphism is square free |
protected IsSemisimple.aeval (p : K[X]) : (aeval f p).IsSemisimple :=
let R := K[X] ⧸ Ideal.span {minpoly K f}
have : Module.Finite K R :=
(AdjoinRoot.powerBasis' <| minpoly.monic <| Algebra.IsIntegral.isIntegral f).finite
have : IsReduced R := (Ideal.isRadical_iff_quotient_reduced _).mp <|
span_minpoly_e... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.aeval | null |
IsSemisimple.of_mem_adjoin_singleton {a : End K M}
(ha : a ∈ Algebra.adjoin K {f}) : a.IsSemisimple := by
rw [Algebra.adjoin_singleton_eq_range_aeval] at ha; obtain ⟨p, rfl⟩ := ha; exact .aeval hf _ | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.of_mem_adjoin_singleton | null |
protected IsSemisimple.pow (n : ℕ) : (f ^ n).IsSemisimple :=
.of_mem_adjoin_singleton hf (pow_mem (Algebra.self_mem_adjoin_singleton _ _) _) | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.pow | null |
IsSemisimple.of_mem_adjoin_pair {a : End K M} (ha : a ∈ Algebra.adjoin K {f, g}) :
a.IsSemisimple := by
let R := K[X] ⧸ Ideal.span {minpoly K f}
let S := AdjoinRoot ((minpoly K g).map <| algebraMap K R)
have : Module.Finite K R :=
(AdjoinRoot.powerBasis' <| minpoly.monic <| Algebra.IsIntegral.isIntegral f... | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.of_mem_adjoin_pair | null |
IsSemisimple.add_of_commute : (f + g).IsSemisimple := .of_mem_adjoin_pair
comm hf hg <| add_mem (Algebra.subset_adjoin <| .inl rfl) (Algebra.subset_adjoin <| .inr rfl) | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.add_of_commute | null |
IsSemisimple.sub_of_commute : (f - g).IsSemisimple := .of_mem_adjoin_pair
comm hf hg <| sub_mem (Algebra.subset_adjoin <| .inl rfl) (Algebra.subset_adjoin <| .inr rfl) | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.sub_of_commute | null |
IsSemisimple.mul_of_commute : (f * g).IsSemisimple := .of_mem_adjoin_pair
comm hf hg <| mul_mem (Algebra.subset_adjoin <| .inl rfl) (Algebra.subset_adjoin <| .inr rfl) | theorem | LinearAlgebra | [
"Mathlib.Algebra.Module.Torsion",
"Mathlib.FieldTheory.Perfect",
"Mathlib.LinearAlgebra.AnnihilatingPolynomial",
"Mathlib.RingTheory.Artinian.Instances",
"Mathlib.RingTheory.Ideal.Quotient.Nilpotent",
"Mathlib.RingTheory.SimpleModule.Basic"
] | Mathlib/LinearAlgebra/Semisimple.lean | IsSemisimple.mul_of_commute | null |
IsOrtho (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x : M₁) (y : M₂) : Prop :=
B x y = 0 | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrtho | The proposition that two elements of a sesquilinear map space are orthogonal |
isOrtho_def {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} {x y} : B.IsOrtho x y ↔ B x y = 0 :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrtho_def | null |
isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x := by
dsimp only [IsOrtho]
rw [map_zero B, zero_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrtho_zero_left | null |
isOrtho_zero_right (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B x (0 : M₂) :=
map_zero (B x) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrtho_zero_right | null |
isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x := by
simp_rw [isOrtho_def, flip_apply]
open scoped Function in -- required for scoped `on` notation | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrtho_flip | null |
IsOrthoᵢ (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) (v : n → M₁) : Prop :=
Pairwise (B.IsOrtho on v) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsOrthoᵢ | A set of vectors `v` is orthogonal with respect to some bilinear map `B` if and only
if for all `i ≠ j`, `B (v i) (v j) = 0`. For orthogonality between two elements, use
`BilinForm.isOrtho` |
isOrthoᵢ_def {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {v : n → M₁} :
B.IsOrthoᵢ v ↔ ∀ i j : n, i ≠ j → B (v i) (v j) = 0 :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrthoᵢ_def | null |
isOrthoᵢ_flip (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) {v : n → M₁} :
B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v := by
simp_rw [isOrthoᵢ_def]
constructor <;> exact fun h i j hij ↦ h j i hij.symm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isOrthoᵢ_flip | null |
ortho_smul_left {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₁} (ha : a ≠ 0) :
IsOrtho B x y ↔ IsOrtho B (a • x) y := by
dsimp only [IsOrtho]
constructor <;> intro H
· rw [map_smulₛₗ₂, H, smul_zero]
· rw [map_smulₛₗ₂, smul_eq_zero] at H
rcases H with H | H
· rw [map_eq_zero I₁] at H
trivial
·... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ortho_smul_left | null |
ortho_smul_right {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₂} {ha : a ≠ 0} :
IsOrtho B x y ↔ IsOrtho B x (a • y) := by
dsimp only [IsOrtho]
constructor <;> intro H
· rw [map_smulₛₗ, H, smul_zero]
· rw [map_smulₛₗ, smul_eq_zero] at H
rcases H with H | H
· simp only [map_eq_zero] at H
exfalso
... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ortho_smul_right | null |
linearIndependent_of_isOrthoᵢ {B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V} {v : n → V₁}
(hv₁ : B.IsOrthoᵢ v) (hv₂ : ∀ i, ¬B.IsOrtho (v i) (v i)) : LinearIndependent K₁ v := by
classical
rw [linearIndependent_iff']
intro s w hs i hi
have : B (s.sum fun i : n ↦ w i • v i) (v i) = 0 := by rw [hs, map_zero, zero_apply]... | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | linearIndependent_of_isOrthoᵢ | A set of orthogonal vectors `v` with respect to some sesquilinear map `B` is linearly
independent if for all `i`, `B (v i) (v i) ≠ 0`. |
IsRefl (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M) : Prop :=
∀ x y, B x y = 0 → B y x = 0 | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsRefl | The proposition that a sesquilinear map is reflexive |
eq_zero : ∀ {x y}, B x y = 0 → B y x = 0 := fun {x y} ↦ H x y | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | eq_zero | null |
eq_iff {x y} : B x y = 0 ↔ B y x = 0 := ⟨H x y, H y x⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | eq_iff | null |
ortho_comm {x y} : IsOrtho B x y ↔ IsOrtho B y x :=
⟨eq_zero H, eq_zero H⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ortho_comm | null |
domRestrict (p : Submodule R₁ M₁) : (B.domRestrict₁₂ p p).IsRefl :=
fun _ _ ↦ by
simp_rw [domRestrict₁₂_apply]
exact H _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | domRestrict | null |
@[simp]
flip_isRefl_iff : B.flip.IsRefl ↔ B.IsRefl :=
⟨fun h x y H ↦ h y x ((B.flip_apply _ _).trans H), fun h x y ↦ h y x⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | flip_isRefl_iff | null |
ker_flip_eq_bot (H : B.IsRefl) (h : LinearMap.ker B = ⊥) : LinearMap.ker B.flip = ⊥ := by
refine ker_eq_bot'.mpr fun _ hx ↦ ker_eq_bot'.mp h _ ?_
ext
exact H _ _ (LinearMap.congr_fun hx _) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ker_flip_eq_bot | null |
ker_eq_bot_iff_ker_flip_eq_bot (H : B.IsRefl) :
LinearMap.ker B = ⊥ ↔ LinearMap.ker B.flip = ⊥ := by
refine ⟨ker_flip_eq_bot H, fun h ↦ ?_⟩
exact (congr_arg _ B.flip_flip.symm).trans (ker_flip_eq_bot (flip_isRefl_iff.mpr H) h) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ker_eq_bot_iff_ker_flip_eq_bot | null |
IsSymm (B : M →ₛₗ[I] M →ₗ[R] R) : Prop where
protected eq : ∀ x y, I (B x y) = B y x | structure | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsSymm | The proposition that a sesquilinear form is symmetric |
isSymm_def {B : M →ₛₗ[I] M →ₗ[R] R} : B.IsSymm ↔ ∀ x y, I (B x y) = B y x :=
⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isSymm_def | null |
isRefl (H : B.IsSymm) : B.IsRefl := fun x y H1 ↦ by
rw [← H.eq]
simp [H1] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isRefl | null |
ortho_comm (H : B.IsSymm) {x y} : IsOrtho B x y ↔ IsOrtho B y x :=
H.isRefl.ortho_comm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ortho_comm | null |
domRestrict (H : B.IsSymm) (p : Submodule R M) : (B.domRestrict₁₂ p p).IsSymm where
eq _ _ := by
simp_rw [domRestrict₁₂_apply]
exact H.eq _ _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | domRestrict | null |
@[simp]
isSymm_zero : (0 : M →ₛₗ[I] M →ₗ[R] R).IsSymm := ⟨fun _ _ => map_zero _⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isSymm_zero | null |
BilinMap.isSymm_iff_eq_flip {N : Type*} [AddCommMonoid N] [Module R N]
{B : LinearMap.BilinMap R M N} : (∀ x y, B x y = B y x) ↔ B = B.flip := by
simp [LinearMap.ext_iff₂] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | BilinMap.isSymm_iff_eq_flip | null |
isSymm_iff_eq_flip {B : LinearMap.BilinForm R M} : B.IsSymm ↔ B = B.flip :=
isSymm_def.trans BilinMap.isSymm_iff_eq_flip | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isSymm_iff_eq_flip | null |
IsAlt (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M) : Prop :=
∀ x, B x x = 0
variable (H : B.IsAlt)
include H | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAlt | The proposition that a sesquilinear map is alternating |
IsAlt.self_eq_zero (x : M₁) : B x x = 0 :=
H x | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAlt.self_eq_zero | null |
IsAlt.eq_of_add_add_eq_zero [IsCancelAdd M] {a b c : M₁} (hAdd : a + b + c = 0) :
B a b = B b c := by
have : B a a + B a b + B a c = B a c + B b c + B c c := by
simp_rw [← map_add, ← map_add₂, hAdd, map_zero, LinearMap.zero_apply]
rw [H, H, zero_add, add_zero, add_comm] at this
exact add_left_cancel this | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | IsAlt.eq_of_add_add_eq_zero | null |
neg (H : B.IsAlt) (x y : M₁) : -B x y = B y x := by
have H1 : B (y + x) (y + x) = 0 := self_eq_zero H (y + x)
simpa [map_add, self_eq_zero H, add_eq_zero_iff_neg_eq] using H1 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | neg | null |
isRefl (H : B.IsAlt) : B.IsRefl := by
intro x y h
rw [← neg H, h, neg_zero] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | isRefl | null |
ortho_comm (H : B.IsAlt) {x y} : IsOrtho B x y ↔ IsOrtho B y x :=
H.isRefl.ortho_comm | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Basis.Basic",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/SesquilinearForm.lean | ortho_comm | null |
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