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Metamath Proof Explorer

Theorem cphdir

Description: Distributive law for inner product (right-distributivity). Equation I3 of Ponnusamy p. 362. Complex version of ipdir . (Contributed by Mario Carneiro, 16-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 cphipcj.h 
 |-  ., = ( .i ` W )
2 cphipcj.v 
 |-  V = ( Base ` W )
3 cphdir.P 
 |-  .+ = ( +g ` W )
4 cphphl 
 |-  ( W e. CPreHil -> W e. PreHil )
5 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
6 eqid 
 |-  ( +g ` ( Scalar ` W ) ) = ( +g ` ( Scalar ` W ) )
7 5 1 2 3 6 ipdir 
 |-  ( ( W e. PreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .+ B ) ., C ) = ( ( A ., C ) ( +g ` ( Scalar ` W ) ) ( B ., C ) ) )
8 4 7 sylan 
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .+ B ) ., C ) = ( ( A ., C ) ( +g ` ( Scalar ` W ) ) ( B ., C ) ) )
9 cphclm 
 |-  ( W e. CPreHil -> W e. CMod )
10 5 clmadd 
 |-  ( W e. CMod -> + = ( +g ` ( Scalar ` W ) ) )
11 9 10 syl 
 |-  ( W e. CPreHil -> + = ( +g ` ( Scalar ` W ) ) )
12 11 adantr 
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> + = ( +g ` ( Scalar ` W ) ) )
13 12 oveqd 
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A ., C ) + ( B ., C ) ) = ( ( A ., C ) ( +g ` ( Scalar ` W ) ) ( B ., C ) ) )
14 8 13 eqtr4d 
 |-  ( ( W e. CPreHil /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( A .+ B ) ., C ) = ( ( A ., C ) + ( B ., C ) ) )