Theorem clmvslinv

Description: Minus a vector plus itself. (Contributed by NM, 4-Dec-2006) (Revised by AV, 28-Sep-2021)

	Ref	Expression
Hypotheses	clmpm1dir.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	clmpm1dir.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
	clmpm1dir.a	⊢ + = ( +g ‘ 𝑊 )
	clmvsrinv.0	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	clmvslinv	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → ( ( - 1 · 𝐴 ) + 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		clmpm1dir.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		clmpm1dir.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
3		clmpm1dir.a	⊢ + = ( +g ‘ 𝑊 )
4		clmvsrinv.0	⊢ 0 = ( 0g ‘ 𝑊 )
5		eqid	⊢ ( invg ‘ 𝑊 ) = ( invg ‘ 𝑊 )
6		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
7	1 5 6 2	clmvneg1	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → ( - 1 · 𝐴 ) = ( ( invg ‘ 𝑊 ) ‘ 𝐴 ) )
8	7	oveq1d	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → ( ( - 1 · 𝐴 ) + 𝐴 ) = ( ( ( invg ‘ 𝑊 ) ‘ 𝐴 ) + 𝐴 ) )
9		clmgrp	⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ Grp )
10	1 3 4 5	grplinv	⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ) → ( ( ( invg ‘ 𝑊 ) ‘ 𝐴 ) + 𝐴 ) = 0 )
11	9 10	sylan	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → ( ( ( invg ‘ 𝑊 ) ‘ 𝐴 ) + 𝐴 ) = 0 )
12	8 11	eqtrd	⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ) → ( ( - 1 · 𝐴 ) + 𝐴 ) = 0 )