anisotropic_of_prod — Mathlib

English

If Q1 and Q2 are positive definite, then their product is positive definite.

Русский

Если Q1 и Q2 положительно определены, то их произведение положительно определено.

LaTeX

$$[PartialOrder P] [AddLeftMono P] implies (Q1.prod Q2).PosDef if Q1.PosDef and Q2.PosDef.$$

Lean4

/-- If a product is anisotropic then its components must be. The converse is not true. -/
theorem anisotropic_of_prod {Q₁ : QuadraticMap R M₁ P} {Q₂ : QuadraticMap R M₂ P} (h : (Q₁.prod Q₂).Anisotropic) :
    Q₁.Anisotropic ∧ Q₂.Anisotropic :=
  by
  simp_rw [Anisotropic, prod_apply, Prod.forall, Prod.mk_eq_zero] at h
  constructor
  · intro x hx
    refine (h x 0 ?_).1
    rw [hx, zero_add, map_zero]
  · intro x hx
    refine (h 0 x ?_).2
    rw [hx, add_zero, map_zero]

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