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
0vfval.5
Assertion
Ref Expression
0vfval GId

Proof of Theorem 0vfval
StepHypRef Expression
1 elex 2964 . 2
2 fo1st 6366 . . . . . . 7
3 fofn 5655 . . . . . . 7
42, 3ax-mp 8 . . . . . 6
5 ssv 3368 . . . . . 6
6 fnco 5553 . . . . . 6
74, 4, 5, 6mp3an 1279 . . . . 5
8 df-va 22074 . . . . . 6
98fneq1i 5539 . . . . 5
107, 9mpbir 201 . . . 4
11 fvco2 5798 . . . 4 GId GId
1210, 11mpan 652 . . 3 GId GId
13 0vfval.5 . . . 4
14 df-0v 22077 . . . . 5 GId
1514fveq1i 5729 . . . 4 GId
1613, 15eqtri 2456 . . 3 GId
17 0vfval.2 . . . 4
1817fveq2i 5731 . . 3 GId GId
1912, 16, 183eqtr4g 2493 . 2 GId
201, 19syl 16 1 GId
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cvv 2956   wss 3320   crn 4879   ccom 4882   wfn 5449  wfo 5452  cfv 5454  c1st 6347  GIdcgi 21775  cpv 22064  cn0v 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