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 ⌀ |
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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), div_nonneg h₂.le h₁₂.le, div_nonneg h₁.le h₁₂.le, ?_, ?_, ?_⟩
· rw [← add_div, add_comm, div_self h₁₂.ne']
· rw [div_eq_inv_mul, mul_smul, smul_add, ← H, ← add_smul, add_comm r₂, inv_smul_smul₀ h₁₂.ne']
· rw [div_eq_inv_mul, mul_smul, smul_add, H, ← add_smul, add_comm r₂, inv_smul_smul₀ h₁₂.ne'] | 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,
LinearMap.sub_apply, LinearMap.coe_comp, LinearMap.smul_apply, smul_eq_mul, smul_sub, sub_smul,
smul_smul, sub_mul, comp_apply, map_sub, map_smul, add_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_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_mem_iff_right hy] using p.smul_mem (f y) hx | 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_ne_zero (α := R) <| by simp [← h]
suffices f y • x ∈ p by
have aux : f y • x ∈ p ⊓ (R ∙ x) := ⟨this, Submodule.mem_span_singleton.mpr ⟨f y, rfl⟩⟩
rw [hp.eq_bot, Submodule.mem_bot, smul_eq_zero] at aux
exact aux.resolve_right hx
specialize h' hy
simp only [Submodule.mem_comap, LinearEquiv.coe_coe, reflection_apply] at h'
simpa using p.sub_mem h' hy
· have hy' : f y = 0 := by simpa using h' hy
simpa [reflection_apply, hy']
/-! ### Powers of the product of two reflections
Let $M$ be a module over a commutative ring $R$. Let $x, y \in M$ and $f, g \in M^*$ with
$f(x) = g(y) = 2$. The corresponding reflections $r_1, r_2 \colon M \to M$ (`Module.reflection`) are
given by $r_1z = z - f(z) x$ and $r_2 z = z - g(z) y$. These are linear automorphisms of $M$.
To define reflection representations of a Coxeter group, it is important to be able to compute the
order of the composition $r_1 r_2$.
Note that if $M$ is a real inner product space and $r_1$ and $r_2$ are both orthogonal
reflections (i.e. $f(z) = 2 \langle x, z \rangle / \langle x, x \rangle$ and
$g(z) = 2 \langle y, z\rangle / \langle y, y\rangle$ for all $z \in M$),
then $r_1 r_2$ is a rotation by the angle
$$\cos^{-1}\left(\frac{f(y) g(x) - 2}{2}\right)$$
and one may determine the order of $r_1 r_2$ accordingly.
However, if $M$ does not have an inner product, and even if $R$ is not $\mathbb{R}$, then we may
instead use the formulas in this section. These formulas all involve evaluating Chebyshev
$S$-polynomials (`Polynomial.Chebyshev.S`) at $t = f(y) g(x) - 2$, and they hold over any
commutative ring. -/ | 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) +
((S R ((m - 1) / 2)).eval t * ((S R (m / 2)).eval t + (S R ((m - 2) / 2)).eval t)) •
((f y * g z - f z) • x - g z • y) := by
induction m with
| zero => simp
| succ m ih =>
/- Now, let us collect two facts about the evaluations of `S r k`. These easily follow from the
properties of the `S` polynomials. -/
have S_eval_t_sub_two (k : ℤ) :
(S R (k - 2)).eval t = t * (S R (k - 1)).eval t - (S R k).eval t := by
simp [S_sub_two]
have S_eval_t_sq_add_S_eval_t_sq (k : ℤ) :
(S R k).eval t ^ 2 + (S R (k + 1)).eval t ^ 2 - t * (S R k).eval t * (S R (k + 1)).eval t
= 1 := by
simpa using congr_arg (Polynomial.eval t) (S_sq_add_S_sq R k)
rw [pow_succ', LinearEquiv.mul_apply, ih, LinearEquiv.mul_apply]
simp only [reflection_apply, map_add, map_sub, map_smul, hf, hg]
push_cast
rw [← Int.mul_ediv_add_emod m 2]
set k : ℤ := m / 2
set e : ℤ := m % 2
simp_rw [add_assoc (2 * k), add_sub_assoc (2 * k), add_comm (2 * k),
add_mul_ediv_left _ k (by simp : (2 : ℤ) ≠ 0)]
have he : e = 0 ∨ e = 1 := by omega
clear_value e
/- Now, equate the coefficients on both sides. These linear combinations were
found using `polyrith`. -/
match_scalars
· rfl
· linear_combination (norm := skip) (-g z * f y * (S R (e - 1 + k)).eval t +
f z * (S R (e - 1 + k)).eval t) * S_eval_t_sub_two (e + k) +
(-g z * f y + f z) * S_eval_t_sq_add_S_eval_t_sq (k - 1)
subst ht
obtain rfl | rfl : e = 0 ∨ e = 1 := he <;> ring_nf
· linear_combination (norm := skip)
g z * (S R (e - 1 + k)).eval t * S_eval_t_sub_two (e + k) +
g z * S_eval_t_sq_add_S_eval_t_sq (k - 1)
subst ht
obtain rfl | rfl : e = 0 ∨ e = 1 := he <;> ring_nf | 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 - g).smulRight y - f.smulRight x) +
((S R ((m - 1) / 2)).eval t * ((S R (m / 2)).eval t + (S R ((m - 2) / 2)).eval t)) •
((f y • g - f).smulRight x - g.smulRight y) := by
ext z
simpa using reflection_mul_reflection_pow_apply hf hg m z 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 | 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) +
((S R ((m - 1) / 2)).eval t * ((S R (m / 2)).eval t + (S R ((m - 2) / 2)).eval t)) •
((f y * g z - f z) • x - g z • y) := by
induction m using Int.negInduction with
| nat m => exact_mod_cast reflection_mul_reflection_pow_apply hf hg m z t ht
| neg _ m =>
have ht' : t = g x * f y - 2 := by rwa [mul_comm (g x)]
rw [zpow_neg, ← inv_zpow, mul_inv_rev, reflection_inv, reflection_inv, zpow_natCast,
reflection_mul_reflection_pow_apply hg hf m z t ht', add_right_comm z]
have aux (a b : ℤ) (hab : a + b = -3 := by omega) : a / 2 = -(b / 2) - 2 := by omega
rw [aux (-m - 3) m, aux (-m - 2) (m - 1), aux (-m - 1) (m - 2), aux (-m) (m - 3)]
simp only [S_neg_sub_two, Polynomial.eval_neg]
ring_nf | 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 - g).smulRight y - f.smulRight x) +
((S R ((m - 1) / 2)).eval t * ((S R (m / 2)).eval t + (S R ((m - 2) / 2)).eval t)) •
((f y • g - f).smulRight x - g.smulRight y) := by
ext z
simpa using reflection_mul_reflection_zpow_apply hf hg m z 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_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_reflection_zpow_apply`, it is
easier to prove it from scratch. -/
have S_eval_t_sub_two (k : ℤ) :
(S R (k - 2)).eval t = (f y * g x - 2) * (S R (k - 1)).eval t - (S R k).eval t := by
simp [S_sub_two, ht]
induction m with
| zero => simp
| succ m ih =>
rw [add_comm (m : ℤ) 1, zpow_one_add, LinearEquiv.mul_apply, LinearEquiv.mul_apply, ih]
simp only [reflection_apply, map_add, map_sub, map_smul, hf, hg]
match_scalars
· linear_combination (norm := ring_nf) -S_eval_t_sub_two (m + 1)
· ring_nf
| pred m ih =>
rw [sub_eq_add_neg (-m : ℤ) 1, add_comm (-m : ℤ) (-1), zpow_add, zpow_neg_one, mul_inv_rev,
reflection_inv, reflection_inv, LinearEquiv.mul_apply, LinearEquiv.mul_apply, ih]
simp only [reflection_apply, map_add, map_sub, map_smul, hf, hg]
match_scalars
· linear_combination (norm := ring_nf) -S_eval_t_sub_two (-m)
· linear_combination (norm := ring_nf) g x * S_eval_t_sub_two (-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_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_mul_reflection_zpow_apply_self hf hg m t ht]
simp only [reflection_apply, map_add, map_smul, hg]
module | 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_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_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; simp at hf₁
let u := reflection hg₁ * reflection hf₁
have hu : u = LinearMap.id (R := R) (M := M) + (f - g).smulRight x := by
ext y
simp only [u, reflection_apply, hg₁, two_smul, LinearEquiv.coe_toLinearMap_mul,
LinearMap.id_coe, LinearEquiv.coe_coe, Module.End.mul_apply, LinearMap.add_apply, id_eq,
LinearMap.coe_smulRight, LinearMap.sub_apply, map_sub, map_smul, sub_add_cancel_left,
smul_neg, sub_neg_eq_add, sub_smul]
abel
replace hu : ∀ (n : ℕ),
↑(u ^ n) = LinearMap.id (R := R) (M := M) + (n : R) • (f - g).smulRight x := by
intro n
induction n with
| zero => simp
| succ n ih =>
have : ((f - g).smulRight x).comp ((n : R) • (f - g).smulRight x) = 0 := by
ext; simp [hf₁, hg₁]
rw [pow_succ', LinearEquiv.coe_toLinearMap_mul, ih, hu, add_mul, mul_add, mul_add]
simp_rw [Module.End.mul_eq_comp, LinearMap.comp_id, LinearMap.id_comp, this, add_zero,
add_assoc, Nat.cast_succ, add_smul, one_smul]
suffices IsOfFinOrder u by
obtain ⟨n, hn₀, hn₁⟩ := isOfFinOrder_iff_pow_eq_one.mp this
replace hn₁ : (↑(u ^ n) : M →ₗ[R] M) = LinearMap.id := LinearEquiv.toLinearMap_inj.mpr hn₁
simpa [hn₁, hn₀.ne', hx, sub_eq_zero] using hu n
exact u.isOfFinOrder_of_finite_of_span_eq_top_of_mapsTo hΦ₁ hΦ₂ (hg₂.comp hf₂) | 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
linear algebra and combinatorics since the finiteness assumption is the key. |
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).subtype.dualMap g := by
set Φ' : Set (span R Φ) := range (inclusion <| Submodule.subset_span (R := R) (s := Φ))
rw [← finite_coe_iff] at hΦ₁
have hΦ'₁ : Φ'.Finite := finite_range (inclusion Submodule.subset_span)
have hΦ'₂ : span R Φ' = ⊤ := by
simp only [Φ']
rw [range_inclusion]
simp
let x' : span R Φ := ⟨x, hx⟩
have this : ∀ {F : Dual R M}, MapsTo (preReflection x F) Φ Φ →
MapsTo (preReflection x' ((span R Φ).subtype.dualMap F)) Φ' Φ' := by
intro F hF ⟨y, hy⟩ hy'
simp only [Φ'] at hy' ⊢
rw [range_inclusion] at hy'
simp only [SetLike.coe_sort_coe, mem_setOf_eq] at hy' ⊢
rw [range_inclusion]
exact hF hy'
exact eq_of_preReflection_mapsTo hΦ'₁ hΦ'₂ hf₁ (this hf₂) hg₁ (this hg₂)
variable {y}
variable {g : Dual R 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 | 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
rw [smul_smul, hgxfy, smul_smul, ← Nat.cast_smul_eq_nsmul R (2 * 2), show 2 * 2 = 4 from rfl,
Nat.cast_ofNat]
simp only [iterate_succ', comp_apply, ih, two_smul, smul_sub, smul_add, map_add,
LinearEquiv.trans_apply, reflection_apply_self, map_neg, reflection_apply, neg_sub, map_sub,
map_nsmul, map_smul, smul_neg, hz, add_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 | 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_range_add_nsmul_iff, sub_ne_zero] | 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
rw [hfy, two_smul R x, two_smul R y, ← two_zsmul, ← two_zsmul] at h
exact smul_right_injective _ two_ne_zero h
rw [← not_infinite] at hΦ
contrapose! hΦ
apply ((infinite_range_reflection_reflection_iterate_iff hfx hgy
(by rw [hfy, hgx]; norm_cast)).mpr hΦ).mono
rw [range_subset_iff]
intro n
rw [← IsFixedPt.image_iterate ((bijOn_reflection_of_mapsTo hfx hxfΦ).comp
(bijOn_reflection_of_mapsTo hgy hygΦ)).image_eq n]
exact mem_image_of_mem _ hyΦ | 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 - ⟨i, rfl⟩ - ⟨j, rfl⟩ hij
congr
suffices ∀ k, c i (r k) = c j (r k) by
rw [← EmbeddingLike.apply_eq_iff_eq r]
exact eq_of_mapsTo_reflection_of_mem (f := c i) (g := c j) hfin (h_two i) (h_two j)
(by rw [← this, h_two]) (by rw [this, h_two]) (h_mapsTo i) (h_mapsTo j) (mem_range_self j)
intro k
simpa using LinearMap.congr_fun hij ⟨r k, Submodule.subset_span (mem_range_self k)⟩ | 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.orderIsoMapComap <| AEval.restrict_equiv_mapSubmodule f p hp).trans
(Submodule.mapIic _)
simp_rw [IsSemisimple, isSemisimpleModule_iff, e.complementedLattice_iff, disjoint_iff,
← (OrderIso.Iic _ _).complementedLattice_iff, Iic.complementedLattice_iff, Subtype.forall,
Subtype.exists, Subtype.mk_le_mk, Sublattice.mk_inf_mk, Sublattice.mk_sup_mk, Subtype.mk.injEq,
exists_and_left, exists_and_right, invtSubmodule.mk_eq_bot_iff, exists_prop, and_assoc]
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_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] using LinearMap.congr_fun he x
simp_rw [IsSemisimple, isSemisimpleModule_iff,
(Submodule.orderIsoMapComap e).complementedLattice_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 | _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
exact eq_zero_of_isNilpotent_isSemisimple hn hs
ext x
obtain ⟨k : ℕ, hk : f ^ k = 0⟩ := hn
let p := Submodule.span R {(f ^ i) x | (i : ℕ) (_ : i ≤ k)}
have hp₁ : p ∈ f.invtSubmodule := by
simp only [mem_invtSubmodule, p, Submodule.span_le]
rintro - ⟨i, hi, rfl⟩
apply Submodule.subset_span
rcases lt_or_eq_of_le hi with hik | rfl
· exact ⟨i + 1, hik, by simpa [Module.End.pow_apply] using iterate_succ_apply' f i x⟩
· exact ⟨i, by simp [hk]⟩
have hp₂ : Module.Finite R p := by
let g : ℕ → M := fun i ↦ (f ^ i) x
have hg : {(f ^ i) x | (i : ℕ) (_ : i ≤ k)} = g '' Iic k := by ext; simp [g]
exact Module.Finite.span_of_finite _ <| hg ▸ toFinite (g '' Iic k)
simpa [LinearMap.restrict_apply, Subtype.ext_iff] using
LinearMap.congr_fun (this p hp₁ hp₂) ⟨x, Submodule.subset_span ⟨0, k.zero_le, rfl⟩⟩
@[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 | 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_mem (h hx) (p.smul_mem μ hx)⟩
simpa using p.add_mem (h hx) (p.smul_mem μ hx) | 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_equiv_mapSubmodule f p hp).trans <|
Submodule.mapIic _
exact (isSemisimpleModule_iff ..).mpr (e.complementedLattice_iff.mpr inferInstance) | 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_top
exact ⟨q, hq₁, hq₂, codisjoint_iff.mpr hq₃⟩
@[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_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 ↦ ⟨fun h x hx ↦ ?_, fun h x hx ↦ p.sub_mem (h hx) (p.smul_mem μ hx)⟩
simpa using p.add_mem (h hx) (p.smul_mem μ hx) | 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.isSemisimple_iff ((LinearMap.restrict f hp).restrict hq₁) (LinearMap.restrict f this) rfl]
exact hf _ this (Finite.map q p.subtype) | 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] ⧸ Ideal.span {p}
have : IsReduced R :=
(Ideal.isRadical_iff_quotient_reduced _).mp (isRadical_iff_span_singleton.mp hp.isRadical)
have : FiniteDimensional K R := (AdjoinRoot.powerBasis hp.ne_zero).finite
have : IsArtinianRing R := .of_finite K R
have : IsSemisimpleRing R := IsArtinianRing.isSemisimpleRing_of_isReduced R
letI : Module R (AEval' f) := Module.IsTorsionBySet.module hpf
let e : AEval' f →ₛₗ[Ideal.Quotient.mk (Ideal.span {p})] AEval' f :=
{ AddMonoidHom.id _ with map_smul' := fun _ _ ↦ rfl }
exact (e.isSemisimpleModule_iff_of_bijective bijective_id).mpr inferInstance
variable [FiniteDimensional K M] | 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_eq_annihilator K f ▸ hf.annihilator_isRadical
isSemisimple_of_squarefree_aeval_eq_zero ((minpoly.isRadical K _).squarefree <|
minpoly.ne_zero <| .of_finite K <| Ideal.Quotient.mkₐ K (.span {minpoly K f}) p) <| by
rw [← Ideal.Quotient.liftₐ_comp (.span {minpoly K f}) (aeval f)
fun a h ↦ by rwa [Ideal.span, ← minpoly.ker_aeval_eq_span_minpoly] at h, aeval_algHom,
AlgHom.comp_apply, AlgHom.comp_apply, ← aeval_algHom_apply, minpoly.aeval, map_zero] | 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).finite
have : Module.Finite R S :=
(AdjoinRoot.powerBasis' <| (minpoly.monic <| Algebra.IsIntegral.isIntegral g).map _).finite
have : Module.Finite K S := .trans R S
have : IsArtinianRing R := .of_finite K R
have : IsReduced R := (Ideal.isRadical_iff_quotient_reduced _).mp <|
span_minpoly_eq_annihilator K f ▸ hf.annihilator_isRadical
have : IsReduced S := by
simp_rw [S, AdjoinRoot, ← Ideal.isRadical_iff_quotient_reduced, ← isRadical_iff_span_singleton]
exact (PerfectField.separable_iff_squarefree.mpr hg.minpoly_squarefree).map.squarefree.isRadical
let φ : S →ₐ[K] End K M := Ideal.Quotient.liftₐ _ (eval₂AlgHom' (Ideal.Quotient.liftₐ _ (aeval f)
fun a ↦ ?_) g ?_) ((Ideal.span_singleton_le_iff_mem _).mpr ?_ : _ ≤ RingHom.ker _)
rotate_left 1
· rw [Ideal.span, ← minpoly.ker_aeval_eq_span_minpoly]; exact id
· rintro ⟨p⟩; exact p.induction_on (fun k ↦ by simp [R, Algebra.commute_algebraMap_left])
(fun p q hp hq ↦ by simpa [R] using hp.add_left hq)
fun n k ↦ by simpa [R, pow_succ, ← mul_assoc _ _ X] using (·.mul_left comm)
· simpa only [RingHom.mem_ker, eval₂AlgHom'_apply, eval₂_map, AlgHom.comp_algebraMap_of_tower]
using minpoly.aeval K g
have : Algebra.adjoin K {f, g} ≤ φ.range := Algebra.adjoin_le fun x ↦ by
rintro (hx | hx) <;> rw [hx]
· exact ⟨AdjoinRoot.of _ (AdjoinRoot.root _), (eval₂_C _ _).trans (aeval_X f)⟩
· exact ⟨AdjoinRoot.root _, eval₂_X _ _⟩
obtain ⟨p, rfl⟩ := (AlgHom.mem_range _).mp (this ha)
refine isSemisimple_of_squarefree_aeval_eq_zero
((minpoly.isRadical K p).squarefree <| minpoly.ne_zero <| .of_finite K p) ?_
rw [aeval_algHom, φ.comp_apply, minpoly.aeval, map_zero] | 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
· exact H | 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
exact ha H
· exact H | 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]
have hsum : (s.sum fun j : n ↦ I₁ (w j) • B (v j) (v i)) = I₁ (w i) • B (v i) (v i) := by
apply Finset.sum_eq_single_of_mem i hi
intro j _hj hij
rw [isOrthoᵢ_def.1 hv₁ _ _ hij, smul_zero]
simp_rw [B.map_sum₂, map_smulₛₗ₂, hsum] at this
apply (map_eq_zero I₁).mp
exact (smul_eq_zero.mp this).elim _root_.id (hv₂ i · |>.elim) | 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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