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Theorem bafval 25201
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 5866 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 5230 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 25193 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5876 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 6718 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5950 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 5254 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2480 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5860 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5860 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 5230 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2534 . . 3  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  ran  ( +v `  U ) )
136, 12pm2.61i 164 . 2  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
14 bafval.1 . 2  |-  X  =  ( BaseSet `  U )
15 bafval.2 . . 3  |-  G  =  ( +v `  U
)
1615rneqi 5229 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2506 1  |-  X  =  ran  G
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ran crn 5000   ` cfv 5588   +vcpv 25182   BaseSetcba 25183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-ba 25193
This theorem is referenced by:  nvi  25211  nvgf  25215  nvsf  25216  nvgcl  25217  nvcom  25218  nvass  25219  nvadd32  25221  nvrcan  25222  nvlcan  25223  nvadd4  25224  nvscl  25225  nvsid  25226  nvsass  25227  nvdi  25229  nvdir  25230  nv2  25231  nvzcl  25233  nv0rid  25234  nv0lid  25235  nv0  25236  nvsz  25237  nvinv  25238  nvinvfval  25239  nvmval  25241  nvmfval  25243  nvnnncan1  25247  nvnnncan2  25248  nvnegneg  25250  nvrinv  25252  nvlinv  25253  nvaddsubass  25257  nvaddsub  25258  nvdm  25268  nvmtri2  25279  cnnvba  25288  sspba  25344  isph  25441  phpar  25443  ip0i  25444  ipdirilem  25448  hhba  25788  hhssabloi  25882  hhshsslem1  25887
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