Theorem abl32

Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015) (Revised by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablcom.p	⊢ + = ( +g ‘ 𝐺 )
	abl32.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
	abl32.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	abl32.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	abl32.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	abl32	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑋 + 𝑍 ) + 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablcom.p	⊢ + = ( +g ‘ 𝐺 )
3		abl32.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
4		abl32.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		abl32.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		abl32.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7		ablcmn	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ CMnd )
8	3 7	syl	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
9	1 2	cmn32	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑋 + 𝑍 ) + 𝑌 ) )
10	8 4 5 6 9	syl13anc	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑋 + 𝑍 ) + 𝑌 ) )