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Theorem bafval 22075
Description: Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
bafval.1  |-  X  =  ( BaseSet `  U )
bafval.2  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
bafval  |-  X  =  ran  G

Proof of Theorem bafval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 5089 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 22067 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5734 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 5125 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5798 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 5119 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2439 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5714 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5714 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 5089 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2493 . . 3  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  ran  ( +v `  U ) )
136, 12pm2.61i 158 . 2  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
14 bafval.1 . 2  |-  X  =  ( BaseSet `  U )
15 bafval.2 . . 3  |-  G  =  ( +v `  U
)
1615rneqi 5088 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2465 1  |-  X  =  ran  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   ran crn 4871   ` cfv 5446   +vcpv 22056   BaseSetcba 22057
This theorem is referenced by:  nvi  22085  nvgf  22089  nvsf  22090  nvgcl  22091  nvcom  22092  nvass  22093  nvadd32  22095  nvrcan  22096  nvlcan  22097  nvadd4  22098  nvscl  22099  nvsid  22100  nvsass  22101  nvdi  22103  nvdir  22104  nv2  22105  nvzcl  22107  nv0rid  22108  nv0lid  22109  nv0  22110  nvsz  22111  nvinv  22112  nvinvfval  22113  nvmval  22115  nvmfval  22117  nvnnncan1  22121  nvnnncan2  22122  nvnegneg  22124  nvrinv  22126  nvlinv  22127  nvaddsubass  22131  nvaddsub  22132  nvdm  22142  nvmtri2  22153  cnnvba  22162  sspba  22218  isph  22315  phpar  22317  ip0i  22318  ipdirilem  22322  hhba  22661  hhssabloi  22754  hhshsslem1  22759
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-ba 22067
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