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Theorem nvdm 24049
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 2443 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvdm.2 . . . . . 6  |-  G  =  ( +v `  U
)
31, 2bafval 23982 . . . . 5  |-  ( BaseSet `  U )  =  ran  G
43eqcomi 2447 . . . 4  |-  ran  G  =  ( BaseSet `  U
)
5 nvdm.6 . . . 4  |-  N  =  ( normCV `  U )
64, 5nvf 24046 . . 3  |-  ( U  e.  NrmCVec  ->  N : ran  G --> RR )
7 fdm 5563 . . 3  |-  ( N : ran  G --> RR  ->  dom 
N  =  ran  G
)
86, 7syl 16 . 2  |-  ( U  e.  NrmCVec  ->  dom  N  =  ran  G )
98eqeq2d 2454 1  |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   dom cdm 4840   ran crn 4841   -->wf 5414   ` cfv 5418   RRcr 9281   NrmCVeccnv 23962   +vcpv 23963   BaseSetcba 23964   normCVcnmcv 23968
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-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-1st 6577  df-2nd 6578  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-nmcv 23978
This theorem is referenced by: (None)
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