Theorem ablsubsub

Description: Law for double subtraction. (Contributed by NM, 7-Apr-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ablsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		ablsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		ablsubsub.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
5		ablsubsub.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ablsubsub.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		ablsubsub.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
9	4 8	syl	⊢ ( 𝜑 → 𝐺 ∈ Grp )
10	1 2 3	grpsubsub	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 − ( 𝑌 − 𝑍 ) ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )
11	9 5 6 7 10	syl13anc	⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 − 𝑍 ) ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )
12	1 2 3	grpaddsubass	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )
13	9 5 7 6 12	syl13anc	⊢ ( 𝜑 → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )
14	1 2 3	abladdsub	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( ( 𝑋 − 𝑌 ) + 𝑍 ) )
15	4 5 7 6 14	syl13anc	⊢ ( 𝜑 → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( ( 𝑋 − 𝑌 ) + 𝑍 ) )
16	11 13 15	3eqtr2d	⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 − 𝑍 ) ) = ( ( 𝑋 − 𝑌 ) + 𝑍 ) )