isoCycles₂_hom_comp_i — Mathlib

English

The homomorphism part of isoCycles₂ composed with the i-map equals iCycles(2) composed with chainsIso₂.

Русский

Гомоморфная часть isoCycles₂, составленная с iCycles(2), эквивалентна iCycles(2) после chainsIso₂.

LaTeX

$$$ (\text{isoCycles}_2 A).hom \circ (shortComplexH2 A).moduleCatLeftHomologyData.i = iCycles A 2 \circ (\text{chainsIso}_2 A).hom$$

Lean4

@[reassoc (attr := simp), elementwise (attr := simp)]
theorem isoCycles₂_hom_comp_i :
    (isoCycles₂ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.i = iCycles A 2 ≫ (chainsIso₂ A).hom := by
  simp [isoCycles₂, iCycles, HomologicalComplex.iCycles, ShortComplex.iCycles]

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