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Theorem 2polssN 30030
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 29474 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
21ad3antrrr 711 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  K  e.  CLat )
3 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  p  e.  X )
4 simpllr 736 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  A )
5 eqid 2388 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 2polss.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atssbase 29406 . . . . . 6  |-  A  C_  ( Base `  K )
84, 7syl6ss 3304 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  ( Base `  K
) )
9 eqid 2388 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
10 eqid 2388 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
115, 9, 10lubel 14477 . . . . 5  |-  ( ( K  e.  CLat  /\  p  e.  X  /\  X  C_  ( Base `  K )
)  ->  p ( le `  K ) ( ( lub `  K
) `  X )
)
122, 3, 8, 11syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  p
( le `  K
) ( ( lub `  K ) `  X
) )
1312ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A
)  ->  ( p  e.  X  ->  p ( le `  K ) ( ( lub `  K
) `  X )
) )
1413ss2rabdv 3368 . 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 3272 . . 3  |-  ( A  i^i  X )  =  { p  e.  A  |  p  e.  X }
16 sseqin2 3504 . . . . 5  |-  ( X 
C_  A  <->  ( A  i^i  X )  =  X )
1716biimpi 187 . . . 4  |-  ( X 
C_  A  ->  ( A  i^i  X )  =  X )
1817adantl 453 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( A  i^i  X
)  =  X )
1915, 18syl5reqr 2435 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  =  { p  e.  A  |  p  e.  X } )
20 eqid 2388 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
21 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
2210, 6, 20, 212polvalN 30029 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) ) )
23 sstr 3300 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
247, 23mpan2 653 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
255, 10clatlubcl 14468 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
261, 24, 25syl2an 464 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
275, 9, 6, 20pmapval 29872 . . . 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 457 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
2922, 28eqtrd 2420 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
3014, 19, 293sstr4d 3335 1  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654    i^i cin 3263    C_ wss 3264   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464   lubclub 14327   CLatccla 14464   Atomscatm 29379   HLchlt 29466   pmapcpmap 29612   _|_
PcpolN 30017
This theorem is referenced by:  polcon2N  30034  pclss2polN  30036  sspmaplubN  30040  paddunN  30042  pnonsingN  30048  osumcllem1N  30071  osumcllem11N  30081  pexmidN  30084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-pmap 29619  df-polarityN 30018
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