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Theorem phop 26442
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  =  ( .sOLD `  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 26439 . . 3  |-  Rel  CPreHil OLD
2 1st2nd 6849 . . 3  |-  ( ( Rel  CPreHil OLD  /\  U  e.  CPreHil
OLD )  ->  U  =  <. ( 1st `  U
) ,  ( 2nd `  U ) >. )
31, 2mpan 674 . 2  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. ( 1st `  U
) ,  ( 2nd `  U ) >. )
4 phop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 26209 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 4188 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 phnv 26438 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
8 eqid 2422 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
98nvvc 26217 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
10 vcrel 26149 . . . . . . 7  |-  Rel  CVecOLD
11 1st2nd 6849 . . . . . . 7  |-  ( ( Rel  CVecOLD  /\  ( 1st `  U )  e. 
CVecOLD )  ->  ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
1210, 11mpan 674 . . . . . 6  |-  ( ( 1st `  U )  e.  CVecOLD  ->  ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
13 phop.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
1413vafval 26205 . . . . . . 7  |-  G  =  ( 1st `  ( 1st `  U ) )
15 phop.4 . . . . . . . 8  |-  S  =  ( .sOLD `  U )
1615smfval 26207 . . . . . . 7  |-  S  =  ( 2nd `  ( 1st `  U ) )
1714, 16opeq12i 4189 . . . . . 6  |-  <. G ,  S >.  =  <. ( 1st `  ( 1st `  U
) ) ,  ( 2nd `  ( 1st `  U ) ) >.
1812, 17syl6eqr 2481 . . . . 5  |-  ( ( 1st `  U )  e.  CVecOLD  ->  ( 1st `  U )  = 
<. G ,  S >. )
197, 9, 183syl 18 . . . 4  |-  ( U  e.  CPreHil OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
2019opeq1d 4190 . . 3  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  N >.  =  <. <. G ,  S >. ,  N >. )
216, 20syl5eqr 2477 . 2  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  ( 2nd `  U
) >.  =  <. <. G ,  S >. ,  N >. )
223, 21eqtrd 2463 1  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868   <.cop 4002   Rel wrel 4854   ` cfv 5597   1stc1st 6801   2ndc2nd 6802   CVecOLDcvc 26147   NrmCVeccnv 26186   +vcpv 26187   .sOLDcns 26189   normCVcnmcv 26192   CPreHil OLDccphlo 26436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-1st 6803  df-2nd 6804  df-vc 26148  df-nv 26194  df-va 26197  df-ba 26198  df-sm 26199  df-0v 26200  df-nmcv 26202  df-ph 26437
This theorem is referenced by:  phpar  26448
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