English
The four-way commutation is an Equivalence; equivalence type aligns with Equiv.prodProdProdComm.
Русский
Четырёхкратная коммутативность образует эквивалентность; соответствие видоизменяется через Equiv.prodProdProdComm.
LaTeX
$$$ (RingEquiv.prodProdProdComm R R' S S') : _ ≃+* _ = Equiv.prodProdProdComm R R' S S'. $$$
Lean4
/-- The product of two nontrivial rings is not a domain -/
theorem false_of_nontrivial_of_product_domain (R S : Type*) [Semiring R] [Semiring S] [IsDomain (R × S)] [Nontrivial R]
[Nontrivial S] : False :=
by
have := NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero (show ((0 : R), (1 : S)) * (1, 0) = 0 by simp)
rw [Prod.mk_eq_zero, Prod.mk_eq_zero] at this
rcases this with (⟨_, h⟩ | ⟨h, _⟩)
· exact zero_ne_one h.symm
· exact zero_ne_one h.symm
