Theorem cmn4

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	cmn4	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablcom.p	⊢ + = ( +g ‘ 𝐺 )
3		simp1	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝐺 ∈ CMnd )
4		cmnmnd	⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
5	3 4	syl	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝐺 ∈ Mnd )
6		simp2l	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
7		simp2r	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
8		simp3l	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
9		simp3r	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → 𝑊 ∈ 𝐵 )
10	1 2	cmncom	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
11	3 7 8 10	syl3anc	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
12	1 2 5 6 7 8 9 11	mnd4g	⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( 𝑍 + 𝑊 ) ) = ( ( 𝑋 + 𝑍 ) + ( 𝑌 + 𝑊 ) ) )