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Theorem 2polatN 30743
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 30173 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2 eqid 2296 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polat.a . . . . 5  |-  A  =  ( Atoms `  K )
4 eqid 2296 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
5 2polat.p . . . . 5  |-  P  =  ( _|_ P `  K )
62, 3, 4, 5polatN 30742 . . . 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 5545 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
9 hlop 30174 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 3atbase 30101 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 2opoccl 30006 . . . . 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 30723 . . . 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 30007 . . . . . 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 5545 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  Q )
)
193, 4pmapat 30574 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2018, 19eqtrd 2328 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
2115, 20eqtrd 2328 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
228, 21eqtrd 2328 1  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   ` cfv 5271   Basecbs 13164   occoc 13232   OPcops 29984   OLcol 29986   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   _|_ PcpolN 30713
This theorem is referenced by:  atpsubclN  30756
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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