Theorem ablnnncan

Description: Cancellation law for group subtraction. ( nnncan analog.) (Contributed by NM, 29-Feb-2008) (Revised by AV, 27-Aug-2021)

	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	ablnnncan	⊢ ( 𝜑 → ( ( 𝑋 − ( 𝑌 − 𝑍 ) ) − 𝑍 ) = ( 𝑋 − 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		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
9	3 8	syl	⊢ ( 𝜑 → 𝐺 ∈ Grp )
10	1 2	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
11	9 5 6 10	syl3anc	⊢ ( 𝜑 → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
12	1 7 2 3 4 11 6	ablsubsub4	⊢ ( 𝜑 → ( ( 𝑋 − ( 𝑌 − 𝑍 ) ) − 𝑍 ) = ( 𝑋 − ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) ) )
13	1 7	ablcom	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑌 − 𝑍 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) )
14	3 11 6 13	syl3anc	⊢ ( 𝜑 → ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) = ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) )
15	1 7 2	ablpncan3	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) = 𝑌 )
16	3 6 5 15	syl12anc	⊢ ( 𝜑 → ( 𝑍 ( +g ‘ 𝐺 ) ( 𝑌 − 𝑍 ) ) = 𝑌 )
17	14 16	eqtrd	⊢ ( 𝜑 → ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) = 𝑌 )
18	17	oveq2d	⊢ ( 𝜑 → ( 𝑋 − ( ( 𝑌 − 𝑍 ) ( +g ‘ 𝐺 ) 𝑍 ) ) = ( 𝑋 − 𝑌 ) )
19	12 18	eqtrd	⊢ ( 𝜑 → ( ( 𝑋 − ( 𝑌 − 𝑍 ) ) − 𝑍 ) = ( 𝑋 − 𝑌 ) )