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Theorem phop 22324
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
phop.2  |-  G  =  ( +v `  U
)
phop.4  |-  S  =  ( .s OLD `  U
)
phop.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
phop  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )

Proof of Theorem phop
StepHypRef Expression
1 phrel 22321 . . 3  |-  Rel  CPreHil OLD
2 1st2nd 6396 . . 3  |-  ( ( Rel  CPreHil OLD  /\  U  e.  CPreHil
OLD )  ->  U  =  <. ( 1st `  U
) ,  ( 2nd `  U ) >. )
31, 2mpan 653 . 2  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. ( 1st `  U
) ,  ( 2nd `  U ) >. )
4 phop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 22091 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 3990 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 phnv 22320 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
8 eqid 2438 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
98nvvc 22099 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
10 vcrel 22031 . . . . . . 7  |-  Rel  CVec OLD
11 1st2nd 6396 . . . . . . 7  |-  ( ( Rel  CVec OLD  /\  ( 1st `  U )  e. 
CVec OLD )  ->  ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
1210, 11mpan 653 . . . . . 6  |-  ( ( 1st `  U )  e.  CVec OLD  ->  ( 1st `  U )  =  <. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
13 phop.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
1413vafval 22087 . . . . . . 7  |-  G  =  ( 1st `  ( 1st `  U ) )
15 phop.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
1615smfval 22089 . . . . . . 7  |-  S  =  ( 2nd `  ( 1st `  U ) )
1714, 16opeq12i 3991 . . . . . 6  |-  <. G ,  S >.  =  <. ( 1st `  ( 1st `  U
) ) ,  ( 2nd `  ( 1st `  U ) ) >.
1812, 17syl6eqr 2488 . . . . 5  |-  ( ( 1st `  U )  e.  CVec OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
197, 9, 183syl 19 . . . 4  |-  ( U  e.  CPreHil OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
2019opeq1d 3992 . . 3  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  N >.  =  <. <. G ,  S >. ,  N >. )
216, 20syl5eqr 2484 . 2  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  ( 2nd `  U
) >.  =  <. <. G ,  S >. ,  N >. )
223, 21eqtrd 2470 1  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   <.cop 3819   Rel wrel 4886   ` cfv 5457   1stc1st 6350   2ndc2nd 6351   CVec OLDcvc 22029   NrmCVeccnv 22068   +vcpv 22069   .s OLDcns 22071   normCVcnmcv 22074   CPreHil OLDccphlo 22318
This theorem is referenced by:  phpar  22330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-1st 6352  df-2nd 6353  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084  df-ph 22319
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