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Theorem 0vfval 25697
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 3115 . 2  |-  ( U  e.  V  ->  U  e.  _V )
2 fo1st 6793 . . . . . . 7  |-  1st : _V -onto-> _V
3 fofn 5779 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 5 . . . . . 6  |-  1st  Fn  _V
5 ssv 3509 . . . . . 6  |-  ran  1st  C_ 
_V
6 fnco 5671 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  1st  Fn  _V  /\  ran  1st  C_  _V )  ->  ( 1st  o.  1st )  Fn 
_V )
74, 4, 5, 6mp3an 1322 . . . . 5  |-  ( 1st 
o.  1st )  Fn  _V
8 df-va 25686 . . . . . 6  |-  +v  =  ( 1st  o.  1st )
98fneq1i 5657 . . . . 5  |-  ( +v  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 209 . . . 4  |-  +v  Fn  _V
11 fvco2 5923 . . . 4  |-  ( ( +v  Fn  _V  /\  U  e.  _V )  ->  ( (GId  o.  +v ) `  U )  =  (GId `  ( +v `  U ) ) )
1210, 11mpan 668 . . 3  |-  ( U  e.  _V  ->  (
(GId  o.  +v ) `  U )  =  (GId
`  ( +v `  U ) ) )
13 0vfval.5 . . . 4  |-  Z  =  ( 0vec `  U
)
14 df-0v 25689 . . . . 5  |-  0vec  =  (GId  o.  +v )
1514fveq1i 5849 . . . 4  |-  ( 0vec `  U )  =  ( (GId  o.  +v ) `  U )
1613, 15eqtri 2483 . . 3  |-  Z  =  ( (GId  o.  +v ) `  U )
17 0vfval.2 . . . 4  |-  G  =  ( +v `  U
)
1817fveq2i 5851 . . 3  |-  (GId `  G )  =  (GId
`  ( +v `  U ) )
1912, 16, 183eqtr4g 2520 . 2  |-  ( U  e.  _V  ->  Z  =  (GId `  G )
)
201, 19syl 16 1  |-  ( U  e.  V  ->  Z  =  (GId `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ran crn 4989    o. ccom 4992    Fn wfn 5565   -onto->wfo 5568   ` cfv 5570   1stc1st 6771  GIdcgi 25387   +vcpv 25676   0veccn0v 25679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-va 25686  df-0v 25689
This theorem is referenced by:  nvi  25705  nvzcl  25727  nv0rid  25728  nv0lid  25729  nv0  25730  nvsz  25731  nvrinv  25746  nvlinv  25747  hh0v  26283
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