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Metamath Proof Explorer

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 ) ) )