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Theorem nvop2 22079
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 22073 . . 3  |-  Rel  NrmCVec
2 1st2nd 6385 . . 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 22078 . . 3  |-  N  =  ( 2nd `  U
)
74, 6opeq12i 3981 . 2  |-  <. W ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
83, 7syl6eqr 2485 1  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   <.cop 3809   Rel wrel 4875   ` cfv 5446   1stc1st 6339   2ndc2nd 6340   NrmCVeccnv 22055   normCVcnmcv 22061
This theorem is referenced by:  nvvop  22080  nvi  22085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342  df-nv 22063  df-nmcv 22071
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