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Theorem pmap0 33748
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 2454 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap0.z . . . 4  |-  .0.  =  ( 0. `  K )
31, 2atl0cl 33287 . . 3  |-  ( K  e.  AtLat  ->  .0.  e.  ( Base `  K )
)
4 eqid 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2454 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 pmap0.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 33740 . . 3  |-  ( ( K  e.  AtLat  /\  .0.  e.  ( Base `  K
) )  ->  ( M `  .0.  )  =  { a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }
)
83, 7mpdan 668 . 2  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  {
a  e.  ( Atoms `  K )  |  a ( le `  K
)  .0.  } )
94, 2, 5atnle0 33293 . . . . 5  |-  ( ( K  e.  AtLat  /\  a  e.  ( Atoms `  K )
)  ->  -.  a
( le `  K
)  .0.  )
109nrexdv 2925 . . . 4  |-  ( K  e.  AtLat  ->  -.  E. a  e.  ( Atoms `  K )
a ( le `  K )  .0.  )
11 rabn0 3766 . . . 4  |-  ( { a  e.  ( Atoms `  K )  |  a ( le `  K
)  .0.  }  =/=  (/)  <->  E. a  e.  ( Atoms `  K ) a ( le `  K )  .0.  )
1210, 11sylnibr 305 . . 3  |-  ( K  e.  AtLat  ->  -.  { a  e.  ( Atoms `  K
)  |  a ( le `  K )  .0.  }  =/=  (/) )
13 nne 2654 . . 3  |-  ( -. 
{ a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }  =/=  (/)  <->  { a  e.  (
Atoms `  K )  |  a ( le `  K )  .0.  }  =  (/) )
1412, 13sylib 196 . 2  |-  ( K  e.  AtLat  ->  { a  e.  ( Atoms `  K )  |  a ( le
`  K )  .0. 
}  =  (/) )
158, 14eqtrd 2495 1  |-  ( K  e.  AtLat  ->  ( M `  .0.  )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   {crab 2803   (/)c0 3746   class class class wbr 4401   ` cfv 5527   Basecbs 14293   lecple 14365   0.cp0 15327   Atomscatm 33247   AtLatcal 33248   pmapcpmap 33480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-poset 15236  df-plt 15248  df-glb 15265  df-p0 15329  df-lat 15336  df-covers 33250  df-ats 33251  df-atl 33282  df-pmap 33487
This theorem is referenced by:  pmapeq0  33749  pmapjat1  33836  pol1N  33893  pnonsingN  33916
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