Theorem rngoa32

Description: The addition operation of a ring is commutative. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ringgcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	ringgcl.2	⊢ 𝑋 = ran 𝐺
Assertion	rngoa32	⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ringgcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		ringgcl.2	⊢ 𝑋 = ran 𝐺
3	1	rngoablo	⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ AbelOp )
4	2	ablo32	⊢ ( ( 𝐺 ∈ AbelOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐵 ) )
5	3 4	sylan	⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( ( 𝐴 𝐺 𝐶 ) 𝐺 𝐵 ) )