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Theorem 2polssN 30726
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 30170 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
21ad3antrrr 710 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  K  e.  CLat )
3 simpr 447 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  p  e.  X )
4 simpllr 735 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  A )
5 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 2polss.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atssbase 30102 . . . . . 6  |-  A  C_  ( Base `  K )
84, 7syl6ss 3204 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  ( Base `  K
) )
9 eqid 2296 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
10 eqid 2296 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
115, 9, 10lubel 14242 . . . . 5  |-  ( ( K  e.  CLat  /\  p  e.  X  /\  X  C_  ( Base `  K )
)  ->  p ( le `  K ) ( ( lub `  K
) `  X )
)
122, 3, 8, 11syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  p
( le `  K
) ( ( lub `  K ) `  X
) )
1312ex 423 . . 3  |-  ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A
)  ->  ( p  e.  X  ->  p ( le `  K ) ( ( lub `  K
) `  X )
) )
1413ss2rabdv 3267 . 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 3173 . . 3  |-  ( A  i^i  X )  =  { p  e.  A  |  p  e.  X }
16 sseqin2 3401 . . . . 5  |-  ( X 
C_  A  <->  ( A  i^i  X )  =  X )
1716biimpi 186 . . . 4  |-  ( X 
C_  A  ->  ( A  i^i  X )  =  X )
1817adantl 452 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( A  i^i  X
)  =  X )
1915, 18syl5reqr 2343 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  =  { p  e.  A  |  p  e.  X } )
20 eqid 2296 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
21 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
2210, 6, 20, 212polvalN 30725 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) ) )
23 sstr 3200 . . . . . 6  |-  ( ( X  C_  A  /\  A  C_  ( Base `  K
) )  ->  X  C_  ( Base `  K
) )
247, 23mpan2 652 . . . . 5  |-  ( X 
C_  A  ->  X  C_  ( Base `  K
) )
255, 10clatlubcl 14233 . . . . 5  |-  ( ( K  e.  CLat  /\  X  C_  ( Base `  K
) )  ->  (
( lub `  K
) `  X )  e.  ( Base `  K
) )
261, 24, 25syl2an 463 . . . 4  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( lub `  K
) `  X )  e.  ( Base `  K
) )
275, 9, 6, 20pmapval 30568 . . . 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 456 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( ( pmap `  K
) `  ( ( lub `  K ) `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
2922, 28eqtrd 2328 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
3014, 19, 293sstr4d 3234 1  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    i^i cin 3164    C_ wss 3165   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   lubclub 14092   CLatccla 14229   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   _|_
PcpolN 30713
This theorem is referenced by:  polcon2N  30730  pclss2polN  30732  sspmaplubN  30736  paddunN  30738  pnonsingN  30744  osumcllem1N  30767  osumcllem11N  30777  pexmidN  30780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-pmap 30315  df-polarityN 30714
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