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Theorem 0vfval 21178
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 2809 . 2  |-  ( U  e.  V  ->  U  e.  _V )
2 fo1st 6155 . . . . . . 7  |-  1st : _V -onto-> _V
3 fofn 5469 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 8 . . . . . 6  |-  1st  Fn  _V
5 ssv 3211 . . . . . 6  |-  ran  1st  C_ 
_V
6 fnco 5368 . . . . . 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 21167 . . . . . 6  |-  +v  =  ( 1st  o.  1st )
98fneq1i 5354 . . . . 5  |-  ( +v  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 200 . . . 4  |-  +v  Fn  _V
11 fvco2 5610 . . . 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 21170 . . . . 5  |-  0vec  =  (GId  o.  +v )
1514fveq1i 5542 . . . 4  |-  ( 0vec `  U )  =  ( (GId  o.  +v ) `  U )
1613, 15eqtri 2316 . . 3  |-  Z  =  ( (GId  o.  +v ) `  U )
17 0vfval.2 . . . 4  |-  G  =  ( +v `  U
)
1817fveq2i 5544 . . 3  |-  (GId `  G )  =  (GId
`  ( +v `  U ) )
1912, 16, 183eqtr4g 2353 . 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ran crn 4706    o. ccom 4709    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   1stc1st 6136  GIdcgi 20870   +vcpv 21157   0veccn0v 21160
This theorem is referenced by:  nvi  21186  nvzcl  21208  nv0rid  21209  nv0lid  21210  nv0  21211  nvsz  21212  nvrinv  21227  nvlinv  21228  hh0v  21763
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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-va 21167  df-0v 21170
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