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Theorem nvdm 21243
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 2296 . . . . . 6  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
2 nvdm.2 . . . . . 6  |-  G  =  ( +v `  U
)
31, 2bafval 21176 . . . . 5  |-  ( BaseSet `  U )  =  ran  G
43eqcomi 2300 . . . 4  |-  ran  G  =  ( BaseSet `  U
)
5 nvdm.6 . . . 4  |-  N  =  ( normCV `  U )
64, 5nvf 21240 . . 3  |-  ( U  e.  NrmCVec  ->  N : ran  G --> RR )
7 fdm 5409 . . 3  |-  ( N : ran  G --> RR  ->  dom 
N  =  ran  G
)
86, 7syl 15 . 2  |-  ( U  e.  NrmCVec  ->  dom  N  =  ran  G )
98eqeq2d 2307 1  |-  ( U  e.  NrmCVec  ->  ( X  =  dom  N  <->  X  =  ran  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271   RRcr 8752   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   normCVcnmcv 21162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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