Theorem grpsubid1

Description: Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015)

	Ref	Expression
Hypotheses	grpsubid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubid.o	⊢ 0 = ( 0g ‘ 𝐺 )
	grpsubid.m	⊢ − = ( -g ‘ 𝐺 )
Assertion	grpsubid1	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 0 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		grpsubid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubid.o	⊢ 0 = ( 0g ‘ 𝐺 )
3		grpsubid.m	⊢ − = ( -g ‘ 𝐺 )
4		id	⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵 )
5	1 2	grpidcl	⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
8	1 6 7 3	grpsubval	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 − 0 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 0 ) ) )
9	4 5 8	syl2anr	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 0 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 0 ) ) )
10	2 7	grpinvid	⊢ ( 𝐺 ∈ Grp → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 )
11	10	adantr	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 )
12	11	oveq2d	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 0 ) ) = ( 𝑋 ( +g ‘ 𝐺 ) 0 ) )
13	1 6 2	grprid	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) 0 ) = 𝑋 )
14	9 12 13	3eqtrd	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 0 ) = 𝑋 )