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Theorem pmapssat 33408
Description: The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)
Hypotheses
Ref Expression
pmapssat.b  |-  B  =  ( Base `  K
)
pmapssat.a  |-  A  =  ( Atoms `  K )
pmapssat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapssat  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  A )

Proof of Theorem pmapssat
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 pmapssat.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2443 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 pmapssat.a . . 3  |-  A  =  ( Atoms `  K )
4 pmapssat.m . . 3  |-  M  =  ( pmap `  K
)
51, 2, 3, 4pmapval 33406 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  =  { p  e.  A  |  p
( le `  K
) X } )
6 ssrab2 3442 . 2  |-  { p  e.  A  |  p
( le `  K
) X }  C_  A
75, 6syl6eqss 3411 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2724    C_ wss 3333   class class class wbr 4297   ` cfv 5423   Basecbs 14179   lecple 14250   Atomscatm 32913   pmapcpmap 33146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-pmap 33153
This theorem is referenced by:  pmapssbaN  33409  pmapglb2N  33420  pmapglb2xN  33421  pmapjoin  33501  pmapjat1  33502  pmapjat2  33503  pmapjlln1  33504  hlmod1i  33505  polpmapN  33561  2pmaplubN  33575  pmapj2N  33578  pmapocjN  33579  polatN  33580  pmapsubclN  33595  ispsubcl2N  33596  pl42lem2N  33629  pl42lem3N  33630
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