isoCycles₁_inv_comp_iCycles — Mathlib

English

The inverse of isoCycles₁ composed with iCycles(1) equals the inverse of shortComplexH1 composed with chainsIso₁.

Русский

Обратное изоморфизма isoCycles₁, умноженное на iCycles(1), равно обратному изоморфлению shortComplexH1, составленному с chainsIso₁.

LaTeX

$$$ (\text{isoCycles}_1 A)^{-1} \circ iCycles A 1 = (shortComplexH1 A).moduleCatLeftHomologyData.i \circ (\text{chainsIso}_1 A).inv$$

Lean4

@[reassoc (attr := simp), elementwise (attr := simp)]
theorem isoCycles₁_inv_comp_iCycles :
    (isoCycles₁ A).inv ≫ iCycles A 1 = (shortComplexH1 A).moduleCatLeftHomologyData.i ≫ (chainsIso₁ A).inv :=
  (CommSq.horiz_inv ⟨isoCycles₁_hom_comp_i A⟩).w

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