Theorem cmn32

Description: Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablcom.p	⊢ + = ( +g ‘ 𝐺 )
Assertion	cmn32	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑋 + 𝑍 ) + 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablcom.p	⊢ + = ( +g ‘ 𝐺 )
3		cmnmnd	⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
4	3	adantr	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐺 ∈ Mnd )
5		simpr1	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
6		simpr2	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7		simpr3	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
8	1 2	cmncom	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
9	8	3adant3r1	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
10	1 2 4 5 6 7 9	mnd32g	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑋 + 𝑍 ) + 𝑌 ) )