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Theorem bafval 26068
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 5881 . . . . 5  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
21rneqd 5082 . . . 4  |-  ( u  =  U  ->  ran  ( +v `  u )  =  ran  ( +v
`  U ) )
3 df-ba 26060 . . . 4  |-  BaseSet  =  ( u  e.  _V  |->  ran  ( +v `  u
) )
4 fvex 5891 . . . . 5  |-  ( +v
`  U )  e. 
_V
54rnex 6741 . . . 4  |-  ran  ( +v `  U )  e. 
_V
62, 3, 5fvmpt 5964 . . 3  |-  ( U  e.  _V  ->  ( BaseSet
`  U )  =  ran  ( +v `  U ) )
7 rn0 5106 . . . . 5  |-  ran  (/)  =  (/)
87eqcomi 2442 . . . 4  |-  (/)  =  ran  (/)
9 fvprc 5875 . . . 4  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  (/) )
10 fvprc 5875 . . . . 5  |-  ( -.  U  e.  _V  ->  ( +v `  U )  =  (/) )
1110rneqd 5082 . . . 4  |-  ( -.  U  e.  _V  ->  ran  ( +v `  U
)  =  ran  (/) )
128, 9, 113eqtr4a 2496 . . 3  |-  ( -.  U  e.  _V  ->  (
BaseSet `  U )  =  ran  ( +v `  U ) )
136, 12pm2.61i 167 . 2  |-  ( BaseSet `  U )  =  ran  ( +v `  U )
14 bafval.1 . 2  |-  X  =  ( BaseSet `  U )
15 bafval.2 . . 3  |-  G  =  ( +v `  U
)
1615rneqi 5081 . 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 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   ran crn 4855   ` cfv 5601   +vcpv 26049   BaseSetcba 26050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ba 26060
This theorem is referenced by:  nvi  26078  nvgf  26082  nvsf  26083  nvgcl  26084  nvcom  26085  nvass  26086  nvadd32  26088  nvrcan  26089  nvlcan  26090  nvadd4  26091  nvscl  26092  nvsid  26093  nvsass  26094  nvdi  26096  nvdir  26097  nv2  26098  nvzcl  26100  nv0rid  26101  nv0lid  26102  nv0  26103  nvsz  26104  nvinv  26105  nvinvfval  26106  nvmval  26108  nvmfval  26110  nvnnncan1  26114  nvnnncan2  26115  nvnegneg  26117  nvrinv  26119  nvlinv  26120  nvaddsubass  26124  nvaddsub  26125  nvdm  26135  nvmtri2  26146  cnnvba  26155  sspba  26211  isph  26308  phpar  26310  ip0i  26311  ipdirilem  26315  hhba  26655  hhssabloi  26748  hhshsslem1  26753
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