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Theorem nvop2 24008
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 24002 . . 3  |-  Rel  NrmCVec
2 1st2nd 6641 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 670 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop2.1 . . 3  |-  W  =  ( 1st `  U
)
5 nvop2.6 . . . 4  |-  N  =  ( normCV `  U )
65nmcvfval 24007 . . 3  |-  N  =  ( 2nd `  U
)
74, 6opeq12i 4085 . 2  |-  <. W ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
83, 7syl6eqr 2493 1  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  N >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3904   Rel wrel 4866   ` cfv 5439   1stc1st 6596   2ndc2nd 6597   NrmCVeccnv 23984   normCVcnmcv 23990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447  df-oprab 6116  df-1st 6598  df-2nd 6599  df-nv 23992  df-nmcv 24000
This theorem is referenced by:  nvvop  24009  nvi  24014
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