Theorem crngocom

Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010)

	Ref	Expression
Hypotheses	crngocom.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	crngocom.2	⊢ 𝐻 = ( 2nd ‘ 𝑅 )
	crngocom.3	⊢ 𝑋 = ran 𝐺
Assertion	crngocom	⊢ ( ( 𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		crngocom.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		crngocom.2	⊢ 𝐻 = ( 2nd ‘ 𝑅 )
3		crngocom.3	⊢ 𝑋 = ran 𝐺
4	1 2 3	iscrngo2	⊢ ( 𝑅 ∈ CRingOps ↔ ( 𝑅 ∈ RingOps ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) ) )
5	4	simprbi	⊢ ( 𝑅 ∈ CRingOps → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) )
6		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐻 𝑦 ) = ( 𝐴 𝐻 𝑦 ) )
7		oveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑦 𝐻 𝑥 ) = ( 𝑦 𝐻 𝐴 ) )
8	6 7	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) ↔ ( 𝐴 𝐻 𝑦 ) = ( 𝑦 𝐻 𝐴 ) ) )
9		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐻 𝑦 ) = ( 𝐴 𝐻 𝐵 ) )
10		oveq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐻 𝐴 ) = ( 𝐵 𝐻 𝐴 ) )
11	9 10	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐻 𝑦 ) = ( 𝑦 𝐻 𝐴 ) ↔ ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) )
12	8 11	rspc2v	⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) )
13	5 12	mpan9	⊢ ( ( 𝑅 ∈ CRingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) )
14	13	3impb	⊢ ( ( 𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) )