Theorem grpaddsubass

Description: Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014)

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

Proof

Step	Hyp	Ref	Expression
1		grpsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		grpsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		simpl	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
5		simpr1	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
6		simpr2	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
8	1 7	grpinvcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
9	8	3ad2antr3	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
10	1 2	grpass	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) )
11	4 5 6 9 10	syl13anc	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) )
12	1 2	grpcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
13	12	3adant3r3	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
14		simpr3	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
15	1 2 7 3	grpsubval	⊢ ( ( ( 𝑋 + 𝑌 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
16	13 14 15	syl2anc	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( ( 𝑋 + 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
17	1 2 7 3	grpsubval	⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 − 𝑍 ) = ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
18	6 14 17	syl2anc	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 − 𝑍 ) = ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
19	18	oveq2d	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 + ( 𝑌 − 𝑍 ) ) = ( 𝑋 + ( 𝑌 + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) )
20	11 16 19	3eqtr4d	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − 𝑍 ) = ( 𝑋 + ( 𝑌 − 𝑍 ) ) )