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Theorem bafval 21176
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 5541 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 4922 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 21168 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5555 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 4958 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5618 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 4952 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2300 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5535 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5535 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 4922 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2354 . . 3  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  ran  ( +v `  U ) )
136, 12pm2.61i 156 . 2  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
14 bafval.1 . 2  |-  X  =  ( BaseSet `  U )
15 bafval.2 . . 3  |-  G  =  ( +v `  U
)
1615rneqi 4921 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2326 1  |-  X  =  ran  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ran crn 4706   ` cfv 5271   +vcpv 21157   BaseSetcba 21158
This theorem is referenced by:  nvi  21186  nvgf  21190  nvsf  21191  nvgcl  21192  nvcom  21193  nvass  21194  nvadd32  21196  nvrcan  21197  nvlcan  21198  nvadd4  21199  nvscl  21200  nvsid  21201  nvsass  21202  nvdi  21204  nvdir  21205  nv2  21206  nvzcl  21208  nv0rid  21209  nv0lid  21210  nv0  21211  nvsz  21212  nvinv  21213  nvinvfval  21214  nvmval  21216  nvmfval  21218  nvnnncan1  21222  nvnnncan2  21223  nvnegneg  21225  nvrinv  21227  nvlinv  21228  nvaddsubass  21232  nvaddsub  21233  nvdm  21243  nvmtri2  21254  cnnvba  21263  sspba  21319  isph  21416  phpar  21418  ip0i  21419  ipdirilem  21423  hhba  21762  hhssabloi  21855  hhshsslem1  21860
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-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-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fv 5279  df-ba 21168
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