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Theorem 0vfval 21162
Description: Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
0vfval.2  |-  G  =  ( +v `  U
)
0vfval.5  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
0vfval  |-  ( U  e.  V  ->  Z  =  (GId `  G )
)

Proof of Theorem 0vfval
StepHypRef Expression
1 elex 2796 . 2  |-  ( U  e.  V  ->  U  e.  _V )
2 fo1st 6139 . . . . . . 7  |-  1st : _V -onto-> _V
3 fofn 5453 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 8 . . . . . 6  |-  1st  Fn  _V
5 ssv 3198 . . . . . 6  |-  ran  1st  C_ 
_V
6 fnco 5352 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  1st  Fn  _V  /\  ran  1st  C_  _V )  ->  ( 1st  o.  1st )  Fn 
_V )
74, 4, 5, 6mp3an 1277 . . . . 5  |-  ( 1st 
o.  1st )  Fn  _V
8 df-va 21151 . . . . . 6  |-  +v  =  ( 1st  o.  1st )
98fneq1i 5338 . . . . 5  |-  ( +v  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 200 . . . 4  |-  +v  Fn  _V
11 fvco2 5594 . . . 4  |-  ( ( +v  Fn  _V  /\  U  e.  _V )  ->  ( (GId  o.  +v ) `  U )  =  (GId `  ( +v `  U ) ) )
1210, 11mpan 651 . . 3  |-  ( U  e.  _V  ->  (
(GId  o.  +v ) `  U )  =  (GId
`  ( +v `  U ) ) )
13 0vfval.5 . . . 4  |-  Z  =  ( 0vec `  U
)
14 df-0v 21154 . . . . 5  |-  0vec  =  (GId  o.  +v )
1514fveq1i 5526 . . . 4  |-  ( 0vec `  U )  =  ( (GId  o.  +v ) `  U )
1613, 15eqtri 2303 . . 3  |-  Z  =  ( (GId  o.  +v ) `  U )
17 0vfval.2 . . . 4  |-  G  =  ( +v `  U
)
1817fveq2i 5528 . . 3  |-  (GId `  G )  =  (GId
`  ( +v `  U ) )
1912, 16, 183eqtr4g 2340 . 2  |-  ( U  e.  _V  ->  Z  =  (GId `  G )
)
201, 19syl 15 1  |-  ( U  e.  V  ->  Z  =  (GId `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ran crn 4690    o. ccom 4693    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255   1stc1st 6120  GIdcgi 20854   +vcpv 21141   0veccn0v 21144
This theorem is referenced by:  nvi  21170  nvzcl  21192  nv0rid  21193  nv0lid  21194  nv0  21195  nvsz  21196  nvrinv  21211  nvlinv  21212  hh0v  21747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-va 21151  df-0v 21154
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