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Theorem 2polssN 30104
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 29548 . . . . . 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 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 2polss.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atssbase 29480 . . . . . 6  |-  A  C_  ( Base `  K )
84, 7syl6ss 3191 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  C_  A )  /\  p  e.  A )  /\  p  e.  X )  ->  X  C_  ( Base `  K
) )
9 eqid 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
10 eqid 2283 . . . . . 6  |-  ( lub `  K )  =  ( lub `  K )
115, 9, 10lubel 14226 . . . . 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 3254 . 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 3160 . . 3  |-  ( A  i^i  X )  =  { p  e.  A  |  p  e.  X }
16 sseqin2 3388 . . . . 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 2330 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  =  { p  e.  A  |  p  e.  X } )
20 eqid 2283 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
21 2polss.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
2210, 6, 20, 212polvalN 30103 . . 3  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( ( pmap `  K
) `  ( ( lub `  K ) `  X ) ) )
23 sstr 3187 . . . . . 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 14217 . . . . 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 29946 . . . 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 2315 . 2  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  { p  e.  A  |  p ( le `  K ) ( ( lub `  K ) `
 X ) } )
3014, 19, 293sstr4d 3221 1  |-  ( ( K  e.  HL  /\  X  C_  A )  ->  X  C_  (  ._|_  `  (  ._|_  `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    i^i cin 3151    C_ wss 3152   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   lubclub 14076   CLatccla 14213   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   _|_
PcpolN 30091
This theorem is referenced by:  polcon2N  30108  pclss2polN  30110  sspmaplubN  30114  paddunN  30116  pnonsingN  30122  osumcllem1N  30145  osumcllem11N  30155  pexmidN  30158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-pmap 29693  df-polarityN 30092
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