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Theorem 2polatN 30192
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 29622 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
2 eqid 2366 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
3 2polat.a . . . . 5  |-  A  =  ( Atoms `  K )
4 eqid 2366 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
5 2polat.p . . . . 5  |-  P  =  ( _|_ P `  K )
62, 3, 4, 5polatN 30191 . . . 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 5636 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) ) )
9 hlop 29623 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2366 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1110, 3atbase 29550 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1210, 2opoccl 29455 . . . . 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 30172 . . . 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 29456 . . . . . 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 5636 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  ( ( pmap `  K
) `  Q )
)
193, 4pmapat 30023 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  Q )  =  { Q } )
2018, 19eqtrd 2398 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
2115, 20eqtrd 2398 . 2  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  (
( pmap `  K ) `  ( ( oc `  K ) `  Q
) ) )  =  { Q } )
228, 21eqtrd 2398 1  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( P `  ( P `  { Q } ) )  =  { Q } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   {csn 3729   ` cfv 5358   Basecbs 13356   occoc 13424   OPcops 29433   OLcol 29435   Atomscatm 29524   HLchlt 29611   pmapcpmap 29757   _|_ PcpolN 30162
This theorem is referenced by:  atpsubclN  30205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-pmap 29764  df-polarityN 30163
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