cyclesIsoSc'_inv_iCycles — Mathlib

English

The inverse opcycles-iso composed with fromOpcycles equals fromOpcycles on the sc' side.

Русский

Обратное изоморфирование opcycles композиционно с fromOpcycles даёт fromOpcycles на стороне sc'.

LaTeX

$$$ (K.opcyclesIsoSc' i j k hi hk).inv \circ K.fromOpcycles j k = (K.sc' i j k).fromOpcycles $$$

Lean4

@[reassoc (attr := simp)]
theorem cyclesIsoSc'_inv_iCycles : (K.cyclesIsoSc' i j k hi hk).inv ≫ K.iCycles j = (K.sc' i j k).iCycles :=
  by
  dsimp [cyclesIsoSc']
  erw [ShortComplex.cyclesMap_i]
  apply comp_id

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