Theorem crngcomd

Description: Multiplication is commutative in a commutative ring. (Contributed by SN, 8-Mar-2025)

	Ref	Expression
Hypotheses	crngcomd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	crngcomd.t	⊢ · = ( .r ‘ 𝑅 )
	crngcomd.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
	crngcomd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	crngcomd.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	crngcomd	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		crngcomd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		crngcomd.t	⊢ · = ( .r ‘ 𝑅 )
3		crngcomd.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
4		crngcomd.1	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		crngcomd.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2	crngcom	⊢ ( ( 𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑋 · 𝑌 ) = ( 𝑌 · 𝑋 ) )