distribMulAction — Mathlib

English

The product M × N inherits a distributive mul action from its factors: (r) · (m, n) distributes over multiplication coordinatewise, and r · 1 = 1.

Русский

Произведение M × N наследует действие умножения от факторов: (r) · (m, n) распределяется по умножению по координатам, и r · 1 = 1.

LaTeX

$$$a \\cdot ((m_1, n_1)(m_2, n_2)) = (a \\cdot m_1, a \\cdot n_1)(a \\cdot m_2, a \\cdot n_2)$$$$

Lean4

instance distribMulAction {R : Type*} [Monoid R] [AddMonoid M] [AddMonoid N] [DistribMulAction R M]
    [DistribMulAction R N] : DistribMulAction R (M × N) :=
  { Prod.mulAction, Prod.distribSMul with }

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