Theorem ablsubadd

Description: Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014)

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

Proof

Step	Hyp	Ref	Expression
1		ablsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		ablsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		ablgrp	⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
5	1 2 3	grpsubadd	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ ( 𝑍 + 𝑌 ) = 𝑋 ) )
6	4 5	sylan	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ ( 𝑍 + 𝑌 ) = 𝑋 ) )
7	1 2	ablcom	⊢ ( ( 𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
8	7	3adant3r1	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 + 𝑍 ) = ( 𝑍 + 𝑌 ) )
9	8	eqeq1d	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑌 + 𝑍 ) = 𝑋 ↔ ( 𝑍 + 𝑌 ) = 𝑋 ) )
10	6 9	bitr4d	⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) = 𝑍 ↔ ( 𝑌 + 𝑍 ) = 𝑋 ) )