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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 = ( 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 ) )