Theorem grpsubid

Description: Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014)

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

Proof

Step	Hyp	Ref	Expression
1		grpsubid.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubid.o	⊢ 0 = ( 0g ‘ 𝐺 )
3		grpsubid.m	⊢ − = ( -g ‘ 𝐺 )
4		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
5		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
6	1 4 5 3	grpsubval	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ) )
7	6	anidms	⊢ ( 𝑋 ∈ 𝐵 → ( 𝑋 − 𝑋 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ) )
8	7	adantl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ) )
9	1 4 2 5	grprinv	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 𝑋 ) ) = 0 )
10	8 9	eqtrd	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = 0 )