opcyclesOpIso_hom_naturality — Mathlib

English

The inverse equality fromOpcycles.op ≫ cyclesOpIso.inv equals toCycles: S.fromOpcycles.op ≫ S.cyclesOpIso.inv = S.op.toCycles.

Русский

Обратное равенство fromOpcycles.op ≫ cyclesOpIso.inv противодействие toCycles: S.fromOpcycles.op ≫ S.cyclesOpIso.inv = S.op.toCycles.

LaTeX

$$S.fromOpcycles.op \circ S.cyclesOpIso.inv = S.op.toCycles$$

Lean4

@[reassoc]
theorem opcyclesOpIso_hom_naturality (φ : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology] :
    opcyclesMap (opMap φ) ≫ (S₁.opcyclesOpIso).hom = S₂.opcyclesOpIso.hom ≫ (cyclesMap φ).op := by
  rw [← cancel_epi S₂.op.pOpcycles, p_opcyclesMap_assoc, opMap_τ₂, op_pOpcycles_opcyclesOpIso_hom,
    op_pOpcycles_opcyclesOpIso_hom_assoc, ← op_comp, ← op_comp, cyclesMap_i]

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