Theorem ablsub32

Description: Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015)

	Ref	Expression
Hypotheses	ablnncan.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablnncan.m	⊢ − = ( -g ‘ 𝐺 )
	ablnncan.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
	ablnncan.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ablnncan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	ablsub32.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	ablsub32	⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑍 ) − 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ablnncan.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablnncan.m	⊢ − = ( -g ‘ 𝐺 )
3		ablnncan.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
4		ablnncan.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		ablnncan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		ablsub32.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
8	1 7	ablcom	⊢ ( ( 𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) )
9	3 5 6 8	syl3anc	⊢ ( 𝜑 → ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) )
10	9	oveq2d	⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) ) = ( 𝑋 − ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) ) )
11	1 7 2 3 4 5 6	ablsubsub4	⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( 𝑋 − ( 𝑌 ( +g ‘ 𝐺 ) 𝑍 ) ) )
12	1 7 2 3 4 6 5	ablsubsub4	⊢ ( 𝜑 → ( ( 𝑋 − 𝑍 ) − 𝑌 ) = ( 𝑋 − ( 𝑍 ( +g ‘ 𝐺 ) 𝑌 ) ) )
13	10 11 12	3eqtr4d	⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑍 ) − 𝑌 ) )