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Metamath Proof Explorer

Theorem cphorthcom

Description: Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom . (Contributed by Mario Carneiro, 16-Oct-2015)

Ref Expression
Hypotheses cphipcj.h 
|- ., = ( .i ` W )
cphipcj.v 
|- V = ( Base ` W )
Assertion cphorthcom 
|- ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( ( A ., B ) = 0 <-> ( B ., A ) = 0 ) )

Proof

Step Hyp Ref Expression
1 cphipcj.h 
 |-  ., = ( .i ` W )
2 cphipcj.v 
 |-  V = ( Base ` W )
3 cphphl 
 |-  ( W e. CPreHil -> W e. PreHil )
4 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
5 eqid 
 |-  ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) )
6 4 1 2 5 iporthcom 
 |-  ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ., B ) = ( 0g ` ( Scalar ` W ) ) <-> ( B ., A ) = ( 0g ` ( Scalar ` W ) ) ) )
7 3 6 syl3an1 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( ( A ., B ) = ( 0g ` ( Scalar ` W ) ) <-> ( B ., A ) = ( 0g ` ( Scalar ` W ) ) ) )
8 cphclm 
 |-  ( W e. CPreHil -> W e. CMod )
9 4 clm0 
 |-  ( W e. CMod -> 0 = ( 0g ` ( Scalar ` W ) ) )
10 8 9 syl 
 |-  ( W e. CPreHil -> 0 = ( 0g ` ( Scalar ` W ) ) )
11 10 3ad2ant1 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> 0 = ( 0g ` ( Scalar ` W ) ) )
12 11 eqeq2d 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( ( A ., B ) = 0 <-> ( A ., B ) = ( 0g ` ( Scalar ` W ) ) ) )
13 11 eqeq2d 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( ( B ., A ) = 0 <-> ( B ., A ) = ( 0g ` ( Scalar ` W ) ) ) )
14 7 12 13 3bitr4d 
 |-  ( ( W e. CPreHil /\ A e. V /\ B e. V ) -> ( ( A ., B ) = 0 <-> ( B ., A ) = 0 ) )