Theorem cringm4

Description: Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024)

	Ref	Expression
Hypotheses	cringm4.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	cringm4.2	⊢ · = ( .r ‘ 𝑅 )
Assertion	cringm4	⊢ ( ( 𝑅 ∈ CRing ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · ( 𝑍 · 𝑊 ) ) = ( ( 𝑋 · 𝑍 ) · ( 𝑌 · 𝑊 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cringm4.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		cringm4.2	⊢ · = ( .r ‘ 𝑅 )
3		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4	3	crngmgp	⊢ ( 𝑅 ∈ CRing → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
5	3 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
6	3 2	mgpplusg	⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
7	5 6	cmn4	⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · ( 𝑍 · 𝑊 ) ) = ( ( 𝑋 · 𝑍 ) · ( 𝑌 · 𝑊 ) ) )
8	4 7	syl3an1	⊢ ( ( 𝑅 ∈ CRing ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · ( 𝑍 · 𝑊 ) ) = ( ( 𝑋 · 𝑍 ) · ( 𝑌 · 𝑊 ) ) )