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Theorem pmap1N 29107
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
StepHypRef Expression
1 eqid 2256 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmap1.u . . . 4  |-  .1.  =  ( 1. `  K )
31, 2op1cl 28526 . . 3  |-  ( K  e.  OP  ->  .1.  e.  ( Base `  K
) )
4 eqid 2256 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 pmap1.a . . . 4  |-  A  =  ( Atoms `  K )
6 pmap1.m . . . 4  |-  M  =  ( pmap `  K
)
71, 4, 5, 6pmapval 29097 . . 3  |-  ( ( K  e.  OP  /\  .1.  e.  ( Base `  K
) )  ->  ( M `  .1.  )  =  { p  e.  A  |  p ( le `  K )  .1.  }
)
83, 7mpdan 652 . 2  |-  ( K  e.  OP  ->  ( M `  .1.  )  =  { p  e.  A  |  p ( le `  K )  .1.  }
)
91, 5atbase 28630 . . . . 5  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
101, 4, 2ople1 28532 . . . . 5  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  p ( le `  K )  .1.  )
119, 10sylan2 462 . . . 4  |-  ( ( K  e.  OP  /\  p  e.  A )  ->  p ( le `  K )  .1.  )
1211ralrimiva 2599 . . 3  |-  ( K  e.  OP  ->  A. p  e.  A  p ( le `  K )  .1.  )
13 rabid2 2690 . . 3  |-  ( A  =  { p  e.  A  |  p ( le `  K )  .1.  }  <->  A. p  e.  A  p ( le `  K )  .1.  )
1412, 13sylibr 205 . 2  |-  ( K  e.  OP  ->  A  =  { p  e.  A  |  p ( le `  K )  .1.  }
)
158, 14eqtr4d 2291 1  |-  ( K  e.  OP  ->  ( M `  .1.  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2520   class class class wbr 3983   ` cfv 4659   Basecbs 13096   lecple 13163   1.cp1 14092   OPcops 28513   Atomscatm 28604   pmapcpmap 28837
This theorem is referenced by:  pmapglb2N  29111  pmapglb2xN  29112
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-iota 6211  df-undef 6250  df-riota 6258  df-lub 14056  df-p1 14094  df-oposet 28517  df-ats 28608  df-pmap 28844
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