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Metamath Proof Explorer

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 )