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Theorem bafval 21933
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 5670 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 5039 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 21925 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5684 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 5075 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5747 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 5069 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2393 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5664 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5664 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 5039 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2447 . . 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 5038 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2419 1  |-  X  =  ran  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2901   (/)c0 3573   ran crn 4821   ` cfv 5396   +vcpv 21914   BaseSetcba 21915
This theorem is referenced by:  nvi  21943  nvgf  21947  nvsf  21948  nvgcl  21949  nvcom  21950  nvass  21951  nvadd32  21953  nvrcan  21954  nvlcan  21955  nvadd4  21956  nvscl  21957  nvsid  21958  nvsass  21959  nvdi  21961  nvdir  21962  nv2  21963  nvzcl  21965  nv0rid  21966  nv0lid  21967  nv0  21968  nvsz  21969  nvinv  21970  nvinvfval  21971  nvmval  21973  nvmfval  21975  nvnnncan1  21979  nvnnncan2  21980  nvnegneg  21982  nvrinv  21984  nvlinv  21985  nvaddsubass  21989  nvaddsub  21990  nvdm  22000  nvmtri2  22011  cnnvba  22020  sspba  22076  isph  22173  phpar  22175  ip0i  22176  ipdirilem  22180  hhba  22519  hhssabloi  22612  hhshsslem1  22617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fv 5404  df-ba 21925
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