Theorem ablsubadd23

Description: Commutative/associative law for addition and subtraction in abelian groups. ( subadd23d analog.) (Contributed by AV, 2-Mar-2025)

	Ref	Expression
Hypotheses	ablsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablsubadd.p	⊢ + = ( +g ‘ 𝐺 )
	ablsubadd.m	⊢ − = ( -g ‘ 𝐺 )
Assertion	ablsubadd23	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ablsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		ablsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		3ancomb	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )
5	4	biimpi	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )
6	1 2 3	abladdsub	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( ( 𝑋 − 𝑌 ) + 𝑍 ) )
7	5 6	sylan2	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( ( 𝑋 − 𝑌 ) + 𝑍 ) )
8		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
9	1 2 3	grpaddsubass	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )
10	8 5 9	syl2an	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑍 ) − 𝑌 ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )
11	7 10	eqtr3d	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑍 − 𝑌 ) ) )