MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0vfval Structured version   Unicode version

Theorem 0vfval 22085
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 2964 . 2  |-  ( U  e.  V  ->  U  e.  _V )
2 fo1st 6366 . . . . . . 7  |-  1st : _V -onto-> _V
3 fofn 5655 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 8 . . . . . 6  |-  1st  Fn  _V
5 ssv 3368 . . . . . 6  |-  ran  1st  C_ 
_V
6 fnco 5553 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  1st  Fn  _V  /\  ran  1st  C_  _V )  ->  ( 1st  o.  1st )  Fn 
_V )
74, 4, 5, 6mp3an 1279 . . . . 5  |-  ( 1st 
o.  1st )  Fn  _V
8 df-va 22074 . . . . . 6  |-  +v  =  ( 1st  o.  1st )
98fneq1i 5539 . . . . 5  |-  ( +v  Fn  _V  <->  ( 1st  o. 
1st )  Fn  _V )
107, 9mpbir 201 . . . 4  |-  +v  Fn  _V
11 fvco2 5798 . . . 4  |-  ( ( +v  Fn  _V  /\  U  e.  _V )  ->  ( (GId  o.  +v ) `  U )  =  (GId `  ( +v `  U ) ) )
1210, 11mpan 652 . . 3  |-  ( U  e.  _V  ->  (
(GId  o.  +v ) `  U )  =  (GId
`  ( +v `  U ) ) )
13 0vfval.5 . . . 4  |-  Z  =  ( 0vec `  U
)
14 df-0v 22077 . . . . 5  |-  0vec  =  (GId  o.  +v )
1514fveq1i 5729 . . . 4  |-  ( 0vec `  U )  =  ( (GId  o.  +v ) `  U )
1613, 15eqtri 2456 . . 3  |-  Z  =  ( (GId  o.  +v ) `  U )
17 0vfval.2 . . . 4  |-  G  =  ( +v `  U
)
1817fveq2i 5731 . . 3  |-  (GId `  G )  =  (GId
`  ( +v `  U ) )
1912, 16, 183eqtr4g 2493 . 2  |-  ( U  e.  _V  ->  Z  =  (GId `  G )
)
201, 19syl 16 1  |-  ( U  e.  V  ->  Z  =  (GId `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ran crn 4879    o. ccom 4882    Fn wfn 5449   -onto->wfo 5452   ` cfv 5454   1stc1st 6347  GIdcgi 21775   +vcpv 22064   0veccn0v 22067
This theorem is referenced by:  nvi  22093  nvzcl  22115  nv0rid  22116  nv0lid  22117  nv0  22118  nvsz  22119  nvrinv  22134  nvlinv  22135  hh0v  22670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-va 22074  df-0v 22077
  Copyright terms: Public domain W3C validator