Theorem grpinvval2

Description: A df-neg -like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypotheses	grpsubcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubcl.m	⊢ − = ( -g ‘ 𝐺 )
	grpinvsub.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
	grpinvval2.z	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grpinvval2	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) = ( 0 − 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		grpsubcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubcl.m	⊢ − = ( -g ‘ 𝐺 )
3		grpinvsub.n	⊢ 𝑁 = ( invg ‘ 𝐺 )
4		grpinvval2.z	⊢ 0 = ( 0g ‘ 𝐺 )
5	1 4	grpidcl	⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7	1 6 3 2	grpsubval	⊢ ( ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 0 − 𝑋 ) = ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) )
8	5 7	sylan	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 − 𝑋 ) = ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) )
9	1 3	grpinvcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 )
10	1 6 4	grplid	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑁 ‘ 𝑋 ) ∈ 𝐵 ) → ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) )
11	9 10	syldan	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) )
12	8 11	eqtr2d	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑋 ) = ( 0 − 𝑋 ) )