Theorem ablpncan3

Description: A cancellation law for Abelian groups. (Contributed by NM, 23-Mar-2015)

	Ref	Expression
Hypotheses	ablsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablsubadd.p	⊢ + = ( +g ‘ 𝐺 )
	ablsubadd.m	⊢ − = ( -g ‘ 𝐺 )
Assertion	ablpncan3	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 + ( 𝑌 − 𝑋 ) ) = 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		ablsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		ablsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		simpl	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 ∈ Abel )
5		simprl	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
6		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
7	6	adantr	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
8		simprr	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
9	1 3	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 − 𝑋 ) ∈ 𝐵 )
10	7 8 5 9	syl3anc	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑌 − 𝑋 ) ∈ 𝐵 )
11	1 2	ablcom	⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑌 − 𝑋 ) ∈ 𝐵 ) → ( 𝑋 + ( 𝑌 − 𝑋 ) ) = ( ( 𝑌 − 𝑋 ) + 𝑋 ) )
12	4 5 10 11	syl3anc	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 + ( 𝑌 − 𝑋 ) ) = ( ( 𝑌 − 𝑋 ) + 𝑋 ) )
13	1 2 3	grpnpcan	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑌 − 𝑋 ) + 𝑋 ) = 𝑌 )
14	7 8 5 13	syl3anc	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑌 − 𝑋 ) + 𝑋 ) = 𝑌 )
15	12 14	eqtrd	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 + ( 𝑌 − 𝑋 ) ) = 𝑌 )