Theorem pzriprnglem2

Description: Lemma 2 for pzriprng : The base set of R is the cartesian product of the integers. (Contributed by AV, 17-Mar-2025)

	Ref	Expression
Hypothesis	pzriprng.r	⊢ 𝑅 = ( ℤring ×s ℤring )
Assertion	pzriprnglem2	⊢ ( Base ‘ 𝑅 ) = ( ℤ × ℤ )

Proof

Step	Hyp	Ref	Expression
1		pzriprng.r	⊢ 𝑅 = ( ℤring ×s ℤring )
2		zringring	⊢ ℤring ∈ Ring
3		zringbas	⊢ ℤ = ( Base ‘ ℤring )
4		id	⊢ ( ℤring ∈ Ring → ℤring ∈ Ring )
5	1 3 3 4 4	xpsbas	⊢ ( ℤring ∈ Ring → ( ℤ × ℤ ) = ( Base ‘ 𝑅 ) )
6	2 5	ax-mp	⊢ ( ℤ × ℤ ) = ( Base ‘ 𝑅 )
7	6	eqcomi	⊢ ( Base ‘ 𝑅 ) = ( ℤ × ℤ )