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Theorem 0vfval 24121
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 3079 . 2  |-  ( U  e.  V  ->  U  e.  _V )
2 fo1st 6698 . . . . . . 7  |-  1st : _V -onto-> _V
3 fofn 5722 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 5 . . . . . 6  |-  1st  Fn  _V
5 ssv 3476 . . . . . 6  |-  ran  1st  C_ 
_V
6 fnco 5619 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  1st  Fn  _V  /\  ran  1st  C_  _V )  ->  ( 1st  o.  1st )  Fn 
_V )
74, 4, 5, 6mp3an 1315 . . . . 5  |-  ( 1st 
o.  1st )  Fn  _V
8 df-va 24110 . . . . . 6  |-  +v  =  ( 1st  o.  1st )
98fneq1i 5605 . . . . 5  |-  ( +v  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 209 . . . 4  |-  +v  Fn  _V
11 fvco2 5867 . . . 4  |-  ( ( +v  Fn  _V  /\  U  e.  _V )  ->  ( (GId  o.  +v ) `  U )  =  (GId `  ( +v `  U ) ) )
1210, 11mpan 670 . . 3  |-  ( U  e.  _V  ->  (
(GId  o.  +v ) `  U )  =  (GId
`  ( +v `  U ) ) )
13 0vfval.5 . . . 4  |-  Z  =  ( 0vec `  U
)
14 df-0v 24113 . . . . 5  |-  0vec  =  (GId  o.  +v )
1514fveq1i 5792 . . . 4  |-  ( 0vec `  U )  =  ( (GId  o.  +v ) `  U )
1613, 15eqtri 2480 . . 3  |-  Z  =  ( (GId  o.  +v ) `  U )
17 0vfval.2 . . . 4  |-  G  =  ( +v `  U
)
1817fveq2i 5794 . . 3  |-  (GId `  G )  =  (GId
`  ( +v `  U ) )
1912, 16, 183eqtr4g 2517 . 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 1370    e. wcel 1758   _Vcvv 3070    C_ wss 3428   ran crn 4941    o. ccom 4944    Fn wfn 5513   -onto->wfo 5516   ` cfv 5518   1stc1st 6677  GIdcgi 23811   +vcpv 24100   0veccn0v 24103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-1st 6679  df-va 24110  df-0v 24113
This theorem is referenced by:  nvi  24129  nvzcl  24151  nv0rid  24152  nv0lid  24153  nv0  24154  nvsz  24155  nvrinv  24170  nvlinv  24171  hh0v  24707
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