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Theorem 2polssN 35782
Description: A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polss.a  |-  A  =  ( Atoms `  K )
2polss.p  |-  ._|_  =  ( _|_P `  K
)
Assertion
Ref Expression
2polssN  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )

Proof of Theorem 2polssN
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 hlclat 35226 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
21ad3antrrr 729 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  K  e.  CLat )
3 simpr 461 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  p  e.  X )
4 simpllr 760 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  A )
5 eqid 2457 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 2polss.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atssbase 35158 . . . . . 6  |-  A  C_  ( Base `  K )
84, 7syl6ss 3511 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  ( Base `  K
) )
9 eqid 2457 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
10 eqid 2457 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
115, 9, 10lubel 15879 . . . . 5  |-  ( ( K  e.  CLat  /\  p  e.  X  /\  X  C_  ( Base `  K )
)  ->  p ( le `  K ) ( ( lub `  K
) `  X )
)
122, 3, 8, 11syl3anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  p
( le `  K
) ( ( lub `  K ) `  X
) )
1312ex 434 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A
)  ->  ( p  e.  X  ->  p ( le `  K ) ( ( lub `  K
) `  X )
) )
1413ss2rabdv 3577 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  { p  e.  A  |  p  e.  X }  C_  { p  e.  A  |  p ( le `  K ) ( ( lub `  K
) `  X ) } )
15 dfin5 3479 . . 3  |-  ( A  i^i  X )  =  { p  e.  A  |  p  e.  X }
16 sseqin2 3713 . . . . 5  |-  ( X 
C_  A  <->  ( A  i^i  X )  =  X )
1716biimpi 194 . . . 4  |-  ( X 
C_  A  ->  ( A  i^i  X )  =  X )
1817adantl 466 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( A  i^i  X
)  =  X )
1915, 18syl5reqr 2513 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  =  { p  e.  A  |  p  e.  X } )
20 eqid 2457 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
21 2polss.p . . . 4  |-  ._|_  =  ( _|_P `  K
)
2210, 6, 20, 212polvalN 35781 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) ) )
23 sstr 3507 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
247, 23mpan2 671 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
255, 10clatlubcl 15869 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
261, 24, 25syl2an 477 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
275, 9, 6, 20pmapval 35624 . . . 4  |-  ( ( K  e.  HL  /\  ( ( lub `  K
) `  X )  e.  ( Base `  K
) )  ->  (
( pmap `  K ) `  ( ( lub `  K
) `  X )
)  =  { p  e.  A  |  p
( le `  K
) ( ( lub `  K ) `  X
) } )
2826, 27syldan 470 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
2922, 28eqtrd 2498 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
3014, 19, 293sstr4d 3542 1  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811    i^i cin 3470    C_ wss 3471   class class class wbr 4456   ` cfv 5594   Basecbs 14644   lecple 14719   lubclub 15698   CLatccla 15864   Atomscatm 35131   HLchlt 35218   pmapcpmap 35364   _|_PcpolN 35769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-undef 7020  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-p1 15797  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-pmap 35371  df-polarityN 35770
This theorem is referenced by:  polcon2N  35786  pclss2polN  35788  sspmaplubN  35792  paddunN  35794  pnonsingN  35800  osumcllem1N  35823  osumcllem11N  35833  pexmidN  35836
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