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Theorem nvop2 21936
Description: A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop2.1  |-  W  =  ( 1st `  U
)
nvop2.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvop2  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  N >. )

Proof of Theorem nvop2
StepHypRef Expression
1 nvrel 21930 . . 3  |-  Rel  NrmCVec
2 1st2nd 6333 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 652 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop2.1 . . 3  |-  W  =  ( 1st `  U
)
5 nvop2.6 . . . 4  |-  N  =  ( normCV `  U )
65nmcvfval 21935 . . 3  |-  N  =  ( 2nd `  U
)
74, 6opeq12i 3932 . 2  |-  <. W ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
83, 7syl6eqr 2438 1  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   <.cop 3761   Rel wrel 4824   ` cfv 5395   1stc1st 6287   2ndc2nd 6288   NrmCVeccnv 21912   normCVcnmcv 21918
This theorem is referenced by:  nvvop  21937  nvi  21942
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fv 5403  df-oprab 6025  df-1st 6289  df-2nd 6290  df-nv 21920  df-nmcv 21928
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