domDomCongrEquiv — Mathlib

English

Swapping two arguments in an alternating map yields a sum that cancels due to antisymmetry.

Русский

При перестановке двух аргументов чередующейся карты суммаractions антипсымметрии равна нулю.

LaTeX

$$map_swap_add (f: AlternatingMap R M N ι) (v: ι → M) (i j: ι) (hij: i ≠ j) : f v + f (v ∘ Equiv.swap i j) = 0$$

Lean4

/-- `AlternatingMap.domDomCongr` as an equivalence.

This is declared separately because it does not work with dot notation. -/
@[simps apply symm_apply]
def domDomCongrEquiv (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃+ M [⋀^ι']→ₗ[R] N
    where
  toFun := domDomCongr σ
  invFun := domDomCongr σ.symm
  left_inv
    f := by
    ext
    simp [Function.comp_def]
  right_inv
    m := by
    ext
    simp [Function.comp_def]
  map_add' := domDomCongr_add σ

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