Metamath Proof Explorer

Theorem cphipcj

Description: Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj . (Contributed by Mario Carneiro, 16-Oct-2015)

Ref Expression
Hypotheses cphipcj.h 
|- ., = ( .i ` W )
cphipcj.v 
|- V = ( Base ` W )
Assertion cphipcj 
|- ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( * ` ( A ., B ) ) = ( B ., A ) )

Proof

Step Hyp Ref Expression
1 cphipcj.h 
 |-  ., = ( .i ` W )
2 cphipcj.v 
 |-  V = ( Base ` W )
3 cphclm 
 |-  ( W e. CPreHil -> W e. CMod )
4 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
5 4 clmcj 
 |-  ( W e. CMod -> * = ( *r ` ( Scalar ` W ) ) )
6 3 5 syl 
 |-  ( W e. CPreHil -> * = ( *r ` ( Scalar ` W ) ) )
7 6 3ad2ant1 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> * = ( *r ` ( Scalar ` W ) ) )
8 7 fveq1d 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( * ` ( A ., B ) ) = ( ( *r ` ( Scalar ` W ) ) ` ( A ., B ) ) )
9 cphphl 
 |-  ( W e. CPreHil -> W e. PreHil )
10 eqid 
 |-  ( *r ` ( Scalar ` W ) ) = ( *r ` ( Scalar ` W ) )
11 4 1 2 10 ipcj 
 |-  ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *r ` ( Scalar ` W ) ) ` ( A ., B ) ) = ( B ., A ) )
12 9 11 syl3an1 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( ( *r ` ( Scalar ` W ) ) ` ( A ., B ) ) = ( B ., A ) )
13 8 12 eqtrd 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( * ` ( A ., B ) ) = ( B ., A ) )