English
Given an index set equivalence e : ι ≃ ι', the reindexed dependent product (Π i, A₁ i) is algebra-isomorphic to the dependent product over the reindexed indices (Π i, A₁ (e.symm i)).
Русский
Заданino эквивалентность индексов e : ι ≃ ι', произведение зависящих от индексов функций удовлетворяет существованию алгебра-изоморфизма: Π_i A₁ i ≃ₐ[R] Π_i A₁ (e.symm i).
LaTeX
$$$$ (\Pi i, A_1 i) \cong_R (\Pi i, A_1 (e^{\mathrm{symm}} i)). $$$$
Lean4
/-- Transport dependent functions through an equivalence of the base space.
This is `Equiv.piCongrLeft'` as an `AlgEquiv`.
-/
def piCongrLeft' {ι' : Type*} (e : ι ≃ ι') : (Π i, A₁ i) ≃ₐ[R] Π i, A₁ (e.symm i)
where
__ := RingEquiv.piCongrLeft' A₁ e
commutes' _ := rfl
