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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 
|- .x. = ( .s ` W )
clmpm1dir.a 
|- .+ = ( +g ` W )
clmvsrinv.0 
|- .0. = ( 0g ` W )
Assertion clmvslinv 
|- ( ( W e. CMod /\ A e. V ) -> ( ( -u 1 .x. A ) .+ A ) = .0. )

Proof

Step Hyp Ref Expression
1 clmpm1dir.v 
 |-  V = ( Base ` W )
2 clmpm1dir.s 
 |-  .x. = ( .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 e. CMod /\ A e. V ) -> ( -u 1 .x. A ) = ( ( invg ` W ) ` A ) )
8 7 oveq1d 
 |-  ( ( W e. CMod /\ A e. V ) -> ( ( -u 1 .x. A ) .+ A ) = ( ( ( invg ` W ) ` A ) .+ A ) )
9 clmgrp 
 |-  ( W e. CMod -> W e. Grp )
10 1 3 4 5 grplinv 
 |-  ( ( W e. Grp /\ A e. V ) -> ( ( ( invg ` W ) ` A ) .+ A ) = .0. )
11 9 10 sylan 
 |-  ( ( W e. CMod /\ A e. V ) -> ( ( ( invg ` W ) ` A ) .+ A ) = .0. )
12 8 11 eqtrd 
 |-  ( ( W e. CMod /\ A e. V ) -> ( ( -u 1 .x. A ) .+ A ) = .0. )