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Theorem bafval 22088
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 5731 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 5100 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 22080 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5745 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 5136 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5809 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 5130 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2442 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5725 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5725 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 5100 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2496 . . 3  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  ran  ( +v `  U ) )
136, 12pm2.61i 159 . 2  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
14 bafval.1 . 2  |-  X  =  ( BaseSet `  U )
15 bafval.2 . . 3  |-  G  =  ( +v `  U
)
1615rneqi 5099 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2468 1  |-  X  =  ran  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   ran crn 4882   ` cfv 5457   +vcpv 22069   BaseSetcba 22070
This theorem is referenced by:  nvi  22098  nvgf  22102  nvsf  22103  nvgcl  22104  nvcom  22105  nvass  22106  nvadd32  22108  nvrcan  22109  nvlcan  22110  nvadd4  22111  nvscl  22112  nvsid  22113  nvsass  22114  nvdi  22116  nvdir  22117  nv2  22118  nvzcl  22120  nv0rid  22121  nv0lid  22122  nv0  22123  nvsz  22124  nvinv  22125  nvinvfval  22126  nvmval  22128  nvmfval  22130  nvnnncan1  22134  nvnnncan2  22135  nvnegneg  22137  nvrinv  22139  nvlinv  22140  nvaddsubass  22144  nvaddsub  22145  nvdm  22155  nvmtri2  22166  cnnvba  22175  sspba  22231  isph  22328  phpar  22330  ip0i  22331  ipdirilem  22335  hhba  22674  hhssabloi  22767  hhshsslem1  22772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ba 22080
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