mul_antipode_lTensor_comul_apply — Mathlib

English

Similarly, the map mul' applied to the lTensor of antipode with comul a equals algebraMap(R,A)(counit a).

Русский

Аналогично, mul' применимо к lTensor(antipode) и comul(a) даёт алгебраMap(counit(a)).

LaTeX

$$$\mathrm{mul}'\,R\,A\bigl((\mathrm{antipode}\,R)\,\mathrm{lTensor}\, (\mathrm{Coalgebra}.comul\ a)\bigr) = \mathrm{algebraMap}\,R\,A\bigl(\mathrm{Coalgebra}.counit\ a\bigr).$$$

Lean4

@[simp]
theorem mul_antipode_lTensor_comul_apply (a : A) :
    LinearMap.mul' R A ((antipode R).lTensor A (Coalgebra.comul a)) = algebraMap R A (Coalgebra.counit a) :=
  LinearMap.congr_fun mul_antipode_lTensor_comul a

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