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Theorem 2polssN 30649
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 30093 . . . . . 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 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 2polss.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atssbase 30025 . . . . . 6  |-  A  C_  ( Base `  K )
84, 7syl6ss 3352 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  ( Base `  K
) )
9 eqid 2435 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
10 eqid 2435 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
115, 9, 10lubel 14541 . . . . 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 3416 . 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 3320 . . 3  |-  ( A  i^i  X )  =  { p  e.  A  |  p  e.  X }
16 sseqin2 3552 . . . . 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 2482 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  =  { p  e.  A  |  p  e.  X } )
20 eqid 2435 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
21 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
2210, 6, 20, 212polvalN 30648 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) ) )
23 sstr 3348 . . . . . 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 14532 . . . . 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 30491 . . . 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 2467 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
3014, 19, 293sstr4d 3383 1  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701    i^i cin 3311    C_ wss 3312   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   lubclub 14391   CLatccla 14528   Atomscatm 29998   HLchlt 30085   pmapcpmap 30231   _|_
PcpolN 30636
This theorem is referenced by:  polcon2N  30653  pclss2polN  30655  sspmaplubN  30659  paddunN  30661  pnonsingN  30667  osumcllem1N  30690  osumcllem11N  30700  pexmidN  30703
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 4693
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-rmo 2705  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-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-pmap 30238  df-polarityN 30637
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