Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmap1N Unicode version

Theorem pmap1N 30403
Description: Value of the projective map of a Hilbert lattice at lattice unit. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmap1.u  |-  .1.  =  ( 1. `  K )
pmap1.a  |-  A  =  ( Atoms `  K )
pmap1.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmap1N  |-  ( K  e.  OP  ->  ( M `  .1.  )  =  A )

Proof of Theorem pmap1N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap1.u . . . 4  |-  .1.  =  ( 1. `  K )
31, 2op1cl 29822 . . 3  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
4 eqid 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 pmap1.a . . . 4  |-  A  =  ( Atoms `  K )
6 pmap1.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 30393 . . 3  |-  ( ( K  e.  OP  /\  .1.  e.  ( Base `  K
) )  ->  ( M `  .1.  )  =  { p  e.  A  |  p ( le `  K )  .1.  }
)
83, 7mpdan 650 . 2  |-  ( K  e.  OP  ->  ( M `  .1.  )  =  { p  e.  A  |  p ( le `  K )  .1.  }
)
91, 5atbase 29926 . . . . 5  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
101, 4, 2ople1 29828 . . . . 5  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  p ( le `  K )  .1.  )
119, 10sylan2 461 . . . 4  |-  ( ( K  e.  OP  /\  p  e.  A )  ->  p ( le `  K )  .1.  )
1211ralrimiva 2781 . . 3  |-  ( K  e.  OP  ->  A. p  e.  A  p ( le `  K )  .1.  )
13 rabid2 2877 . . 3  |-  ( A  =  { p  e.  A  |  p ( le `  K )  .1.  }  <->  A. p  e.  A  p ( le `  K )  .1.  )
1412, 13sylibr 204 . 2  |-  ( K  e.  OP  ->  A  =  { p  e.  A  |  p ( le `  K )  .1.  }
)
158, 14eqtr4d 2470 1  |-  ( K  e.  OP  ->  ( M `  .1.  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   class class class wbr 4204   ` cfv 5445   Basecbs 13457   lecple 13524   1.cp1 14455   OPcops 29809   Atomscatm 29900   pmapcpmap 30133
This theorem is referenced by:  pmapglb2N  30407  pmapglb2xN  30408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-undef 6534  df-riota 6540  df-lub 14419  df-p1 14457  df-oposet 29813  df-ats 29904  df-pmap 30140
  Copyright terms: Public domain W3C validator