Theorem ablpnpcan

Description: Cancellation law for mixed addition and subtraction. ( pnpcan analog.) (Contributed by NM, 29-May-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 )
	ablpnpcan.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
	ablpnpcan.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ablpnpcan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	ablpnpcan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion	ablpnpcan	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( 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		ablpnpcan.g	⊢ ( 𝜑 → 𝐺 ∈ Abel )
9		ablpnpcan.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
10		ablpnpcan.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
11		ablpnpcan.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
12	1 2 3	ablsub4	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( ( 𝑋 − 𝑋 ) + ( 𝑌 − 𝑍 ) ) )
13	4 5 6 5 7 12	syl122anc	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( ( 𝑋 − 𝑋 ) + ( 𝑌 − 𝑍 ) ) )
14		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
15	4 14	syl	⊢ ( 𝜑 → 𝐺 ∈ Grp )
16		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
17	1 16 3	grpsubid	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = ( 0g ‘ 𝐺 ) )
18	15 5 17	syl2anc	⊢ ( 𝜑 → ( 𝑋 − 𝑋 ) = ( 0g ‘ 𝐺 ) )
19	18	oveq1d	⊢ ( 𝜑 → ( ( 𝑋 − 𝑋 ) + ( 𝑌 − 𝑍 ) ) = ( ( 0g ‘ 𝐺 ) + ( 𝑌 − 𝑍 ) ) )
20	1 3	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
21	15 6 7 20	syl3anc	⊢ ( 𝜑 → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
22	1 2 16	grplid	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑌 − 𝑍 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) + ( 𝑌 − 𝑍 ) ) = ( 𝑌 − 𝑍 ) )
23	15 21 22	syl2anc	⊢ ( 𝜑 → ( ( 0g ‘ 𝐺 ) + ( 𝑌 − 𝑍 ) ) = ( 𝑌 − 𝑍 ) )
24	13 19 23	3eqtrd	⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( 𝑌 − 𝑍 ) )