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Theorem bafval 23933
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 5686 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 5062 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 23925 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5696 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 6507 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5769 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 5086 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2442 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5680 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5680 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 5062 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2496 . . 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 5061 . 2  |-  ran  G  =  ran  ( +v `  U )
1713, 14, 163eqtr4i 2468 1  |-  X  =  ran  G
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2967   (/)c0 3632   ran crn 4836   ` cfv 5413   +vcpv 23914   BaseSetcba 23915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fv 5421  df-ba 23925
This theorem is referenced by:  nvi  23943  nvgf  23947  nvsf  23948  nvgcl  23949  nvcom  23950  nvass  23951  nvadd32  23953  nvrcan  23954  nvlcan  23955  nvadd4  23956  nvscl  23957  nvsid  23958  nvsass  23959  nvdi  23961  nvdir  23962  nv2  23963  nvzcl  23965  nv0rid  23966  nv0lid  23967  nv0  23968  nvsz  23969  nvinv  23970  nvinvfval  23971  nvmval  23973  nvmfval  23975  nvnnncan1  23979  nvnnncan2  23980  nvnegneg  23982  nvrinv  23984  nvlinv  23985  nvaddsubass  23989  nvaddsub  23990  nvdm  24000  nvmtri2  24011  cnnvba  24020  sspba  24076  isph  24173  phpar  24175  ip0i  24176  ipdirilem  24180  hhba  24520  hhssabloi  24614  hhshsslem1  24619
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