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Theorem pmap0 29084
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
StepHypRef Expression
1 eqid 2256 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap0.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2atl0cl 28623 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
4 eqid 2256 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2256 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 pmap0.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 29076 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  ( Base `  K
) )  ->  ( M `  .0.  )  =  { a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }
)
83, 7mpdan 652 . 2  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  {
a  e.  ( Atoms `  K )  |  a ( le `  K
)  .0.  } )
94, 2, 5atnle0 28629 . . . . 5  |-  ( ( K  e.  AtLat  /\  a  e.  ( Atoms `  K )
)  ->  -.  a
( le `  K
)  .0.  )
109nrexdv 2617 . . . 4  |-  ( K  e.  AtLat  ->  -.  E. a  e.  ( Atoms `  K )
a ( le `  K )  .0.  )
11 rabn0 3416 . . . 4  |-  ( { a  e.  ( Atoms `  K )  |  a ( le `  K
)  .0.  }  =/=  (/)  <->  E. a  e.  ( Atoms `  K ) a ( le `  K )  .0.  )
1210, 11sylnibr 298 . . 3  |-  ( K  e.  AtLat  ->  -.  { a  e.  ( Atoms `  K
)  |  a ( le `  K )  .0.  }  =/=  (/) )
13 nne 2423 . . 3  |-  ( -. 
{ a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }  =/=  (/)  <->  { a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }  =  (/) )
1412, 13sylib 190 . 2  |-  ( K  e.  AtLat  ->  { a  e.  ( Atoms `  K )  |  a ( le
`  K )  .0. 
}  =  (/) )
158, 14eqtrd 2288 1  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   {crab 2519   (/)c0 3397   class class class wbr 3963   ` cfv 4638   Basecbs 13075   lecple 13142   0.cp0 14070   Atomscatm 28583   AtLatcal 28584   pmapcpmap 28816
This theorem is referenced by:  pmapeq0  29085  pmapjat1  29172  pol1N  29229  pnonsingN  29252
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-poset 14007  df-plt 14019  df-lat 14079  df-covers 28586  df-ats 28587  df-atl 28618  df-pmap 28823
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