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Theorem 2polatN 30426
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 29856 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2 eqid 2412 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polat.a . . . . 5  |-  A  =  ( Atoms `  K )
4 eqid 2412 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
5 2polat.p . . . . 5  |-  P  =  ( _|_ P `  K )
62, 3, 4, 5polatN 30425 . . . 4  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  Q ) ) )
71, 6sylan 458 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  Q ) ) )
87fveq2d 5699 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
9 hlop 29857 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2412 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 3atbase 29784 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 2opoccl 29689 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  Q
)  e.  ( Base `  K ) )
139, 11, 12syl2an 464 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( oc `  K ) `  Q
)  e.  ( Base `  K ) )
1410, 2, 4, 5polpmapN 30406 . . . 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 457 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
1610, 2opococ 29690 . . . . . 6  |-  ( ( K  e.  OP  /\  Q  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  (
( oc `  K
) `  Q )
)  =  Q )
179, 11, 16syl2an 464 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( oc `  K ) `  (
( oc `  K
) `  Q )
)  =  Q )
1817fveq2d 5699 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  Q )
)
193, 4pmapat 30257 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2018, 19eqtrd 2444 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
2115, 20eqtrd 2444 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
228, 21eqtrd 2444 1  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3782   ` cfv 5421   Basecbs 13432   occoc 13500   OPcops 29667   OLcol 29669   Atomscatm 29758   HLchlt 29845   pmapcpmap 29991   _|_ PcpolN 30396
This theorem is referenced by:  atpsubclN  30439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-pmap 29998  df-polarityN 30397
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