opposite_mid — Mathlib

English

There exists SMulCommClass M Nᵐᵒᵖ N under the given assumptions, with smul_comm m n z := mul_smul_comm m n z.

Русский

Существуют SMulCommClass M N^{op} N при данных предпосылках, с отношением smul_comm m n z := mul_smul_comm m n z.

LaTeX

$$$SMulCommClass M N^{op} N$ with smul_comm x y z := by applying mul_smul_comm.$$

Lean4

@[to_additive]
instance opposite_mid {M N} [Mul N] [SMul M N] [IsScalarTower M N N] : SMulCommClass M Nᵐᵒᵖ N where
  smul_comm x y
    z := by
    induction y using MulOpposite.rec'
    simp only [smul_mul_assoc, MulOpposite.smul_eq_mul_unop]
      -- The above instance does not create an unwanted diamond, the two paths to
      -- `MulAction αᵐᵒᵖ αᵐᵒᵖ` are defeq.

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