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Theorem nvop 22007
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop.2  |-  G  =  ( +v `  U
)
nvop.4  |-  S  =  ( .s OLD `  U
)
nvop.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvop  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )

Proof of Theorem nvop
StepHypRef Expression
1 nvrel 21922 . . 3  |-  Rel  NrmCVec
2 1st2nd 6325 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 652 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 21927 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 3923 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 eqid 2380 . . . . 5  |-  ( 1st `  U )  =  ( 1st `  U )
8 nvop.2 . . . . 5  |-  G  =  ( +v `  U
)
9 nvop.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
107, 8, 9nvvop 21929 . . . 4  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
1110opeq1d 3925 . . 3  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
126, 11syl5eqr 2426 . 2  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  =  <. <. G ,  S >. ,  N >. )
133, 12eqtrd 2412 1  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   <.cop 3753   Rel wrel 4816   ` cfv 5387   1stc1st 6279   2ndc2nd 6280   NrmCVeccnv 21904   +vcpv 21905   .s OLDcns 21907   normCVcnmcv 21910
This theorem is referenced by:  sspval  22063  isph  22164  hilhhi  22507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fo 5393  df-fv 5395  df-oprab 6017  df-1st 6281  df-2nd 6282  df-vc 21866  df-nv 21912  df-va 21915  df-sm 21917  df-nmcv 21920
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