mul_left — Mathlib

English

If both a commutes with c and b commutes with c, then (a*b) commutes with c.

Русский

Если a коммутирует с c и b коммутирует с c, то (a*b) коммутирует с c.

LaTeX

$$$\\forall S [Semigroup S],\\ a,b,c \\in S,\\ Commute a c \\rightarrow Commute b c \\rightarrow Commute (a b) c$$$

Lean4

/-- If both `a` and `b` commute with `c`, then their product commutes with `c`. -/
@[to_additive (attr := simp) /-- If both `a` and `b` commute with `c`, then their product commutes with `c`. -/
    ]
theorem mul_left (hac : Commute a c) (hbc : Commute b c) : Commute (a * b) c :=
  SemiconjBy.mul_left hac hbc

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