isoCycles₂_inv_comp_iCycles — Mathlib

English

The inverse of isoCycles₂ composed with iCycles(2) equals shortComplexH2.inv composed with chainsIso₂.inv.

Русский

Обратное изоморфизма isoCycles₂, композиция с iCycles(2), равно короткому комплексному инверсу shortComplexH2, композиция с chainsIso₂.inv.

LaTeX

$$$ (\text{isoCycles}_2 A)^{-1} \circ iCycles A 2 = (\text{shortComplexH2 A}).moduleCatLeftHomologyData.i \circ (\text{chainsIso}_2 A).inv$$

Lean4

@[reassoc (attr := simp), elementwise (attr := simp)]
theorem isoCycles₂_inv_comp_iCycles :
    (isoCycles₂ A).inv ≫ iCycles A 2 = (shortComplexH2 A).moduleCatLeftHomologyData.i ≫ (chainsIso₂ A).inv :=
  (CommSq.horiz_inv ⟨isoCycles₂_hom_comp_i A⟩).w

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