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Theorem phop 21510
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 21507 . . 3  |-  Rel  CPreHil OLD
2 1st2nd 6253 . . 3  |-  ( ( Rel  CPreHil OLD  /\  U  e.  CPreHil
OLD )  ->  U  =  <. ( 1st `  U
) ,  ( 2nd `  U ) >. )
31, 2mpan 651 . 2  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. ( 1st `  U
) ,  ( 2nd `  U ) >. )
4 phop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 21277 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 3881 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 phnv 21506 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
8 eqid 2358 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
98nvvc 21285 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
10 vcrel 21217 . . . . . . 7  |-  Rel  CVec OLD
11 1st2nd 6253 . . . . . . 7  |-  ( ( Rel  CVec OLD  /\  ( 1st `  U )  e. 
CVec OLD )  ->  ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
1210, 11mpan 651 . . . . . 6  |-  ( ( 1st `  U )  e.  CVec OLD  ->  ( 1st `  U )  =  <. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
13 phop.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
1413vafval 21273 . . . . . . 7  |-  G  =  ( 1st `  ( 1st `  U ) )
15 phop.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
1615smfval 21275 . . . . . . 7  |-  S  =  ( 2nd `  ( 1st `  U ) )
1714, 16opeq12i 3882 . . . . . 6  |-  <. G ,  S >.  =  <. ( 1st `  ( 1st `  U
) ) ,  ( 2nd `  ( 1st `  U ) ) >.
1812, 17syl6eqr 2408 . . . . 5  |-  ( ( 1st `  U )  e.  CVec OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
197, 9, 183syl 18 . . . 4  |-  ( U  e.  CPreHil OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
2019opeq1d 3883 . . 3  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  N >.  =  <. <. G ,  S >. ,  N >. )
216, 20syl5eqr 2404 . 2  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  ( 2nd `  U
) >.  =  <. <. G ,  S >. ,  N >. )
223, 21eqtrd 2390 1  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   <.cop 3719   Rel wrel 4776   ` cfv 5337   1stc1st 6207   2ndc2nd 6208   CVec OLDcvc 21215   NrmCVeccnv 21254   +vcpv 21255   .s OLDcns 21257   normCVcnmcv 21260   CPreHil OLDccphlo 21504
This theorem is referenced by:  phpar  21516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-1st 6209  df-2nd 6210  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-nmcv 21270  df-ph 21505
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