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Metamath Proof Explorer

Theorem nvop

Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007) (New usage is discouraged.)

Ref Expression
Hypotheses nvop.2 
|- G = ( +v ` U )
nvop.4 
|- S = ( .sOLD ` U )
nvop.6 
|- N = ( normCV ` U )
Assertion nvop 
|- ( U e. NrmCVec -> U = <. <. G , S >. , N >. )

Proof

Step Hyp Ref Expression
1 nvop.2 
 |-  G = ( +v ` U )
2 nvop.4 
 |-  S = ( .sOLD ` U )
3 nvop.6 
 |-  N = ( normCV ` U )
4 nvrel 
 |-  Rel NrmCVec
5 1st2nd 
 |-  ( ( Rel NrmCVec /\ U e. NrmCVec ) -> U = <. ( 1st ` U ) , ( 2nd ` U ) >. )
6 4 5 mpan 
 |-  ( U e. NrmCVec -> U = <. ( 1st ` U ) , ( 2nd ` U ) >. )
7 3 nmcvfval 
 |-  N = ( 2nd ` U )
8 7 opeq2i 
 |-  <. ( 1st ` U ) , N >. = <. ( 1st ` U ) , ( 2nd ` U ) >.
9 eqid 
 |-  ( 1st ` U ) = ( 1st ` U )
10 9 1 2 nvvop 
 |-  ( U e. NrmCVec -> ( 1st ` U ) = <. G , S >. )
11 10 opeq1d 
 |-  ( U e. NrmCVec -> <. ( 1st ` U ) , N >. = <. <. G , S >. , N >. )
12 8 11 eqtr3id 
 |-  ( U e. NrmCVec -> <. ( 1st ` U ) , ( 2nd ` U ) >. = <. <. G , S >. , N >. )
13 6 12 eqtrd 
 |-  ( U e. NrmCVec -> U = <. <. G , S >. , N >. )