comp_prod_eq_prod_comp — Mathlib

English

A second articulation of pi_lift showing compatibility between the quotient lift and pi of paths.

Русский

Повторное изложение pi_lift для совместимости лефта кванто Pi.

LaTeX

$$$\text{pi\_lift} (\gamma) = \langle \mathrm{Path.pi}\; \gamma \rangle$$$

Lean4

/-- Products commute with path composition.
This is `trans_prod_eq_prod_trans` descended to the quotient. -/
theorem comp_prod_eq_prod_comp : prod q₁ q₂ ⬝ prod r₁ r₂ = prod (q₁ ⬝ r₁) (q₂ ⬝ r₂) :=
  by
  induction q₁, q₂ using Quotient.inductionOn₂
  induction r₁, r₂ using Quotient.inductionOn₂
  simp only [prod_lift, ← Path.Homotopic.comp_lift, Path.trans_prod_eq_prod_trans]

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