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 >. )