Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2polatN Structured version   Unicode version

Theorem 2polatN 30630
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 30060 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2 eqid 2435 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polat.a . . . . 5  |-  A  =  ( Atoms `  K )
4 eqid 2435 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
5 2polat.p . . . . 5  |-  P  =  ( _|_ P `  K )
62, 3, 4, 5polatN 30629 . . . 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 5724 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
9 hlop 30061 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 3atbase 29988 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 2opoccl 29893 . . . . 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 30610 . . . 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 29894 . . . . . 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 5724 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  Q )
)
193, 4pmapat 30461 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2018, 19eqtrd 2467 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
2115, 20eqtrd 2467 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
228, 21eqtrd 2467 1  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   ` cfv 5446   Basecbs 13459   occoc 13527   OPcops 29871   OLcol 29873   Atomscatm 29962   HLchlt 30049   pmapcpmap 30195   _|_ PcpolN 30600
This theorem is referenced by:  atpsubclN  30643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-pmap 30202  df-polarityN 30601
  Copyright terms: Public domain W3C validator