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Metamath Proof Explorer

Theorem cphip0l

Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of Ponnusamy p. 361. Complex version of ip0l . (Contributed by Mario Carneiro, 16-Oct-2015)

Ref Expression
Hypotheses cphipcj.h 
|- ., = ( .i ` W )
cphipcj.v 
|- V = ( Base ` W )
cphip0l.z 
|- .0. = ( 0g ` W )
Assertion cphip0l 
|- ( ( W e. CPreHil /\ A e. V ) -> ( .0. ., A ) = 0 )

Proof

Step Hyp Ref Expression
1 cphipcj.h 
 |-  ., = ( .i ` W )
2 cphipcj.v 
 |-  V = ( Base ` W )
3 cphip0l.z 
 |-  .0. = ( 0g ` W )
4 cphphl 
 |-  ( W e. CPreHil -> W e. PreHil )
5 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
6 eqid 
 |-  ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) )
7 5 1 2 6 3 ip0l 
 |-  ( ( W e. PreHil /\ A e. V ) -> ( .0. ., A ) = ( 0g ` ( Scalar ` W ) ) )
8 4 7 sylan 
 |-  ( ( W e. CPreHil /\ A e. V ) -> ( .0. ., A ) = ( 0g ` ( Scalar ` W ) ) )
9 cphclm 
 |-  ( W e. CPreHil -> W e. CMod )
10 5 clm0 
 |-  ( W e. CMod -> 0 = ( 0g ` ( Scalar ` W ) ) )
11 9 10 syl 
 |-  ( W e. CPreHil -> 0 = ( 0g ` ( Scalar ` W ) ) )
12 11 adantr 
 |-  ( ( W e. CPreHil /\ A e. V ) -> 0 = ( 0g ` ( Scalar ` W ) ) )
13 8 12 eqtr4d 
 |-  ( ( W e. CPreHil /\ A e. V ) -> ( .0. ., A ) = 0 )