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Theorem pmap0 29954
Description: Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
pmap0.z  |-  .0.  =  ( 0. `  K )
pmap0.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmap0  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )

Proof of Theorem pmap0
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap0.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2atl0cl 29493 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
4 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2283 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 pmap0.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 29946 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  ( Base `  K
) )  ->  ( M `  .0.  )  =  { a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }
)
83, 7mpdan 649 . 2  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  {
a  e.  ( Atoms `  K )  |  a ( le `  K
)  .0.  } )
94, 2, 5atnle0 29499 . . . . 5  |-  ( ( K  e.  AtLat  /\  a  e.  ( Atoms `  K )
)  ->  -.  a
( le `  K
)  .0.  )
109nrexdv 2646 . . . 4  |-  ( K  e.  AtLat  ->  -.  E. a  e.  ( Atoms `  K )
a ( le `  K )  .0.  )
11 rabn0 3474 . . . 4  |-  ( { a  e.  ( Atoms `  K )  |  a ( le `  K
)  .0.  }  =/=  (/)  <->  E. a  e.  ( Atoms `  K ) a ( le `  K )  .0.  )
1210, 11sylnibr 296 . . 3  |-  ( K  e.  AtLat  ->  -.  { a  e.  ( Atoms `  K
)  |  a ( le `  K )  .0.  }  =/=  (/) )
13 nne 2450 . . 3  |-  ( -. 
{ a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }  =/=  (/)  <->  { a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }  =  (/) )
1412, 13sylib 188 . 2  |-  ( K  e.  AtLat  ->  { a  e.  ( Atoms `  K )  |  a ( le
`  K )  .0. 
}  =  (/) )
158, 14eqtrd 2315 1  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   Atomscatm 29453   AtLatcal 29454   pmapcpmap 29686
This theorem is referenced by:  pmapeq0  29955  pmapjat1  30042  pol1N  30099  pnonsingN  30122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-pmap 29693
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