Theorem grpsubinv

Description: Subtraction of an inverse. (Contributed by NM, 7-Apr-2015)

	Ref	Expression
Hypotheses	grpsubinv.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubinv.p	⊢ + = ( +g ‘ 𝐺 )
	grpsubinv.m	⊢ − = ( -g ‘ 𝐺 )
	grpsubinv.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
	grpsubinv.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	grpsubinv.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	grpsubinv.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	grpsubinv	⊢ ( 𝜑 → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		grpsubinv.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubinv.p	⊢ + = ( +g ‘ 𝐺 )
3		grpsubinv.m	⊢ − = ( -g ‘ 𝐺 )
4		grpsubinv.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
5		grpsubinv.g	⊢ ( 𝜑 → 𝐺 ∈ Grp )
6		grpsubinv.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
7		grpsubinv.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
8	1 4	grpinvcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 )
9	5 7 8	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 )
10	1 2 4 3	grpsubval	⊢ ( ( 𝑋 ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ) )
11	6 9 10	syl2anc	⊢ ( 𝜑 → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ) )
12	1 4	grpinvinv	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
13	5 7 12	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 )
14	13	oveq2d	⊢ ( 𝜑 → ( 𝑋 + ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ) = ( 𝑋 + 𝑌 ) )
15	11 14	eqtrd	⊢ ( 𝜑 → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + 𝑌 ) )