Metamath Proof Explorer


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 = ( 1st𝑅 )
crngocom.2 𝐻 = ( 2nd𝑅 )
crngocom.3 𝑋 = ran 𝐺
Assertion crngocom ( ( 𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) )

Proof

Step Hyp Ref Expression
1 crngocom.1 𝐺 = ( 1st𝑅 )
2 crngocom.2 𝐻 = ( 2nd𝑅 )
3 crngocom.3 𝑋 = ran 𝐺
4 1 2 3 iscrngo2 ( 𝑅 ∈ CRingOps ↔ ( 𝑅 ∈ RingOps ∧ ∀ 𝑥𝑋𝑦𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) ) )
5 4 simprbi ( 𝑅 ∈ CRingOps → ∀ 𝑥𝑋𝑦𝑋 ( 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 ( ( 𝐴𝑋𝐵𝑋 ) → ( ∀ 𝑥𝑋𝑦𝑋 ( 𝑥 𝐻 𝑦 ) = ( 𝑦 𝐻 𝑥 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) ) )
13 5 12 mpan9 ( ( 𝑅 ∈ CRingOps ∧ ( 𝐴𝑋𝐵𝑋 ) ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) )
14 13 3impb ( ( 𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋 ) → ( 𝐴 𝐻 𝐵 ) = ( 𝐵 𝐻 𝐴 ) )