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Theorem nvop 21259
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 21174 . . 3  |-  Rel  NrmCVec
2 1st2nd 6182 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 651 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 21179 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 3816 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 eqid 2296 . . . . 5  |-  ( 1st `  U )  =  ( 1st `  U )
8 nvop.2 . . . . 5  |-  G  =  ( +v `  U
)
9 nvop.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
107, 8, 9nvvop 21181 . . . 4  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
1110opeq1d 3818 . . 3  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
126, 11syl5eqr 2342 . 2  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  =  <. <. G ,  S >. ,  N >. )
133, 12eqtrd 2328 1  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   <.cop 3656   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   NrmCVeccnv 21156   +vcpv 21157   .s OLDcns 21159   normCVcnmcv 21162
This theorem is referenced by:  sspval  21315  isph  21416  hilhhi  21759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-sm 21169  df-nmcv 21172
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