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Theorem pmap0 30576
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 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap0.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2atl0cl 30115 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
4 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2296 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 pmap0.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 30568 . . 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 30121 . . . . 5  |-  ( ( K  e.  AtLat  /\  a  e.  ( Atoms `  K )
)  ->  -.  a
( le `  K
)  .0.  )
109nrexdv 2659 . . . 4  |-  ( K  e.  AtLat  ->  -.  E. a  e.  ( Atoms `  K )
a ( le `  K )  .0.  )
11 rabn0 3487 . . . 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 2463 . . 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 2328 1  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   (/)c0 3468   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   Atomscatm 30075   AtLatcal 30076   pmapcpmap 30308
This theorem is referenced by:  pmapeq0  30577  pmapjat1  30664  pol1N  30721  pnonsingN  30744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-poset 14096  df-plt 14108  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-pmap 30315
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