prodCongr — Mathlib

English

The skew swap map (x,y) ↦ (-y,x) is a linear equivalence between M × N and N × M.

Русский

Перекосный обмен (x,y) ↦ (-y,x) образует линейный эквивалент между M × N и N × M.

LaTeX

$$$(M \times N) \cong_R (N \times M)$ via $(x,y) \mapsto (-y,x)$$$

Lean4

/-- Product of linear equivalences; the maps come from `Equiv.prodCongr`. -/
protected def prodCongr : (M × M₃) ≃ₗ[R] M₂ × M₄ :=
  { e₁.toAddEquiv.prodCongr e₂.toAddEquiv with
    map_smul' := fun c _x => Prod.ext (e₁.map_smulₛₗ c _) (e₂.map_smulₛₗ c _) }

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