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Theorem nvdm 26135
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 2429 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvdm.2 . . . . . 6  |-  G  =  ( +v `  U
)
31, 2bafval 26068 . . . . 5  |-  ( BaseSet `  U )  =  ran  G
43eqcomi 2442 . . . 4  |-  ran  G  =  ( BaseSet `  U
)
5 nvdm.6 . . . 4  |-  N  =  ( normCV `  U )
64, 5nvf 26132 . . 3  |-  ( U  e.  NrmCVec  ->  N : ran  G --> RR )
7 fdm 5750 . . 3  |-  ( N : ran  G --> RR  ->  dom 
N  =  ran  G
)
86, 7syl 17 . 2  |-  ( U  e.  NrmCVec  ->  dom  N  =  ran  G )
98eqeq2d 2443 1  |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870   dom cdm 4854   ran crn 4855   -->wf 5597   ` cfv 5601   RRcr 9537   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   normCVcnmcv 26054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-1st 6807  df-2nd 6808  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064
This theorem is referenced by: (None)
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