mul — Mathlib

English

The product of star projections is a star projection if they commute.

Русский

Произведение звездных проекций является звездной проекцией, если они commute.

LaTeX

$$IsStarProjection(p) ∧ IsStarProjection(q) ∧ Commute(p, q) ⇒ IsStarProjection(p q)$$

Lean4

/-- The product of star projections is a star projection if they commute. -/
theorem mul [NonUnitalSemiring R] [StarRing R] (hp : IsStarProjection p) (hq : IsStarProjection q) (hpq : Commute p q) :
    IsStarProjection (p * q)
    where
  isSelfAdjoint := (IsSelfAdjoint.commute_iff hp.isSelfAdjoint hq.isSelfAdjoint).mp hpq
  isIdempotentElem := hp.isIdempotentElem.mul_of_commute hpq hq.isIdempotentElem

Похожие утверждения