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Theorem 2polatN 30121
Description: Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polat.a  |-  A  =  ( Atoms `  K )
2polat.p  |-  P  =  ( _|_ P `  K )
Assertion
Ref Expression
2polatN  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )

Proof of Theorem 2polatN
StepHypRef Expression
1 hlol 29551 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2 eqid 2283 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polat.a . . . . 5  |-  A  =  ( Atoms `  K )
4 eqid 2283 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
5 2polat.p . . . . 5  |-  P  =  ( _|_ P `  K )
62, 3, 4, 5polatN 30120 . . . 4  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  Q ) ) )
71, 6sylan 457 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  Q ) ) )
87fveq2d 5529 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
9 hlop 29552 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 3atbase 29479 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 2opoccl 29384 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  Q
)  e.  ( Base `  K ) )
139, 11, 12syl2an 463 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( oc `  K ) `  Q
)  e.  ( Base `  K ) )
1410, 2, 4, 5polpmapN 30101 . . . 4  |-  ( ( K  e.  HL  /\  ( ( oc `  K ) `  Q
)  e.  ( Base `  K ) )  -> 
( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
1513, 14syldan 456 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
1610, 2opococ 29385 . . . . . 6  |-  ( ( K  e.  OP  /\  Q  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  Q )
)  =  Q )
179, 11, 16syl2an 463 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  Q )
)  =  Q )
1817fveq2d 5529 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  Q )
)
193, 4pmapat 29952 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2018, 19eqtrd 2315 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
2115, 20eqtrd 2315 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
228, 21eqtrd 2315 1  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   ` cfv 5255   Basecbs 13148   occoc 13216   OPcops 29362   OLcol 29364   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   _|_ PcpolN 30091
This theorem is referenced by:  atpsubclN  30134
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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