auxGroup_indep — Mathlib

English

A culmination of Eckmann–Hilton style argument showing independence of i and j for auxGroup.

Русский

Завершающий результат аргумента по Экенману–Хильтону: независимость i и j для auxGroup.

LaTeX

$$$(\\forall i j, (auxGroup i) = (auxGroup j))$$$

Lean4

theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j :=
  by
  by_cases h : i = j; · rw [h]
  refine Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) ?_)
  rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩
  change Quotient.mk' _ = _
  apply congr_arg Quotient.mk'
  simp only [fromLoop_trans_toLoop, transAt_distrib h, coe_toEquiv, loopHomeo_apply, coe_symm_toEquiv,
    loopHomeo_symm_apply]

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