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Theorem pmap0 29121
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 2258 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap0.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2atl0cl 28660 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
4 eqid 2258 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2258 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 pmap0.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 29113 . . 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 28666 . . . . 5  |-  ( ( K  e.  AtLat  /\  a  e.  ( Atoms `  K )
)  ->  -.  a
( le `  K
)  .0.  )
109nrexdv 2621 . . . 4  |-  ( K  e.  AtLat  ->  -.  E. a  e.  ( Atoms `  K )
a ( le `  K )  .0.  )
11 rabn0 3449 . . . 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 2425 . . 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 2290 1  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   {crab 2522   (/)c0 3430   class class class wbr 3997   ` cfv 4673   Basecbs 13110   lecple 13177   0.cp0 14105   Atomscatm 28620   AtLatcal 28621   pmapcpmap 28853
This theorem is referenced by:  pmapeq0  29122  pmapjat1  29209  pol1N  29266  pnonsingN  29289
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-poset 14042  df-plt 14054  df-lat 14114  df-covers 28623  df-ats 28624  df-atl 28655  df-pmap 28860
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