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Metamath Proof Explorer

Theorem clmpm1dir

Description: Subtractive distributive law for the scalar product of a subcomplex module. (Contributed by NM, 31-Jul-2007) (Revised by AV, 21-Sep-2021)

Ref Expression
Hypotheses clmpm1dir.v 
|- V = ( Base ` W )
clmpm1dir.s 
|- .x. = ( .s ` W )
clmpm1dir.a 
|- .+ = ( +g ` W )
clmpm1dir.k 
|- K = ( Base ` ( Scalar ` W ) )
Assertion clmpm1dir 
|- ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( ( A - B ) .x. C ) = ( ( A .x. C ) .+ ( -u 1 .x. ( B .x. C ) ) ) )

Proof

Step Hyp Ref Expression
1 clmpm1dir.v 
 |-  V = ( Base ` W )
2 clmpm1dir.s 
 |-  .x. = ( .s ` W )
3 clmpm1dir.a 
 |-  .+ = ( +g ` W )
4 clmpm1dir.k 
 |-  K = ( Base ` ( Scalar ` W ) )
5 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
6 eqid 
 |-  ( -g ` W ) = ( -g ` W )
7 simpl 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> W e. CMod )
8 simpr1 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> A e. K )
9 simpr2 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> B e. K )
10 simpr3 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> C e. V )
11 1 2 5 4 6 7 8 9 10 clmsubdir 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( ( A - B ) .x. C ) = ( ( A .x. C ) ( -g ` W ) ( B .x. C ) ) )
12 1 5 2 4 clmvscl 
 |-  ( ( W e. CMod /\ A e. K /\ C e. V ) -> ( A .x. C ) e. V )
13 7 8 10 12 syl3anc 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( A .x. C ) e. V )
14 1 5 2 4 clmvscl 
 |-  ( ( W e. CMod /\ B e. K /\ C e. V ) -> ( B .x. C ) e. V )
15 7 9 10 14 syl3anc 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( B .x. C ) e. V )
16 eqid 
 |-  ( invg ` W ) = ( invg ` W )
17 1 3 16 6 grpsubval 
 |-  ( ( ( A .x. C ) e. V /\ ( B .x. C ) e. V ) -> ( ( A .x. C ) ( -g ` W ) ( B .x. C ) ) = ( ( A .x. C ) .+ ( ( invg ` W ) ` ( B .x. C ) ) ) )
18 13 15 17 syl2anc 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( ( A .x. C ) ( -g ` W ) ( B .x. C ) ) = ( ( A .x. C ) .+ ( ( invg ` W ) ` ( B .x. C ) ) ) )
19 1 16 5 2 clmvneg1 
 |-  ( ( W e. CMod /\ ( B .x. C ) e. V ) -> ( -u 1 .x. ( B .x. C ) ) = ( ( invg ` W ) ` ( B .x. C ) ) )
20 19 eqcomd 
 |-  ( ( W e. CMod /\ ( B .x. C ) e. V ) -> ( ( invg ` W ) ` ( B .x. C ) ) = ( -u 1 .x. ( B .x. C ) ) )
21 7 15 20 syl2anc 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( ( invg ` W ) ` ( B .x. C ) ) = ( -u 1 .x. ( B .x. C ) ) )
22 21 oveq2d 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( ( A .x. C ) .+ ( ( invg ` W ) ` ( B .x. C ) ) ) = ( ( A .x. C ) .+ ( -u 1 .x. ( B .x. C ) ) ) )
23 11 18 22 3eqtrd 
 |-  ( ( W e. CMod /\ ( A e. K /\ B e. K /\ C e. V ) ) -> ( ( A - B ) .x. C ) = ( ( A .x. C ) .+ ( -u 1 .x. ( B .x. C ) ) ) )