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Theorem nvdm 21227
Description: Two ways to express the set of vectors in a normed complex vector space. (Contributed by NM, 31-Jan-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvdm.2  |-  G  =  ( +v `  U
)
nvdm.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvdm  |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )

Proof of Theorem nvdm
StepHypRef Expression
1 eqid 2283 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvdm.2 . . . . . 6  |-  G  =  ( +v `  U
)
31, 2bafval 21160 . . . . 5  |-  ( BaseSet `  U )  =  ran  G
43eqcomi 2287 . . . 4  |-  ran  G  =  ( BaseSet `  U
)
5 nvdm.6 . . . 4  |-  N  =  ( normCV `  U )
64, 5nvf 21224 . . 3  |-  ( U  e.  NrmCVec  ->  N : ran  G --> RR )
7 fdm 5393 . . 3  |-  ( N : ran  G --> RR  ->  dom 
N  =  ran  G
)
86, 7syl 15 . 2  |-  ( U  e.  NrmCVec  ->  dom  N  =  ran  G )
98eqeq2d 2294 1  |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255   RRcr 8736   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   normCVcnmcv 21146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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