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Theorem polatN 30742
Description: The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polat.o  |-  ._|_  =  ( oc `  K )
polat.a  |-  A  =  ( Atoms `  K )
polat.m  |-  M  =  ( pmap `  K
)
polat.p  |-  P  =  ( _|_ P `  K )
Assertion
Ref Expression
polatN  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( M `  (  ._|_  `  Q ) ) )

Proof of Theorem polatN
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 snssi 3775 . . 3  |-  ( Q  e.  A  ->  { Q }  C_  A )
2 polat.o . . . 4  |-  ._|_  =  ( oc `  K )
3 polat.a . . . 4  |-  A  =  ( Atoms `  K )
4 polat.m . . . 4  |-  M  =  ( pmap `  K
)
5 polat.p . . . 4  |-  P  =  ( _|_ P `  K )
62, 3, 4, 5polvalN 30716 . . 3  |-  ( ( K  e.  OL  /\  { Q }  C_  A
)  ->  ( P `  { Q } )  =  ( A  i^i  |^|_
p  e.  { Q }  ( M `  (  ._|_  `  p )
) ) )
71, 6sylan2 460 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( A  i^i  |^|_ p  e.  { Q }  ( M `  (  ._|_  `  p ) ) ) )
8 fveq2 5541 . . . . . 6  |-  ( p  =  Q  ->  (  ._|_  `  p )  =  (  ._|_  `  Q ) )
98fveq2d 5545 . . . . 5  |-  ( p  =  Q  ->  ( M `  (  ._|_  `  p ) )  =  ( M `  (  ._|_  `  Q ) ) )
109iinxsng 3994 . . . 4  |-  ( Q  e.  A  ->  |^|_ p  e.  { Q }  ( M `  (  ._|_  `  p ) )  =  ( M `  (  ._|_  `  Q ) ) )
1110adantl 452 . . 3  |-  ( ( K  e.  OL  /\  Q  e.  A )  -> 
|^|_ p  e.  { Q }  ( M `  (  ._|_  `  p )
)  =  ( M `
 (  ._|_  `  Q
) ) )
1211ineq2d 3383 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( A  i^i  |^|_ p  e.  { Q } 
( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  ( M `  (  ._|_  `  Q )
) ) )
13 olop 30026 . . . . 5  |-  ( K  e.  OL  ->  K  e.  OP )
14 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1514, 3atbase 30101 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 2opoccl 30006 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  ( Base `  K ) )  -> 
(  ._|_  `  Q )  e.  ( Base `  K
) )
1713, 15, 16syl2an 463 . . . 4  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  (  ._|_  `  Q )  e.  ( Base `  K
) )
1814, 3, 4pmapssat 30570 . . . 4  |-  ( ( K  e.  OL  /\  (  ._|_  `  Q )  e.  ( Base `  K
) )  ->  ( M `  (  ._|_  `  Q ) )  C_  A )
1917, 18syldan 456 . . 3  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( M `  (  ._|_  `  Q ) ) 
C_  A )
20 sseqin2 3401 . . 3  |-  ( ( M `  (  ._|_  `  Q ) )  C_  A 
<->  ( A  i^i  ( M `  (  ._|_  `  Q ) ) )  =  ( M `  (  ._|_  `  Q )
) )
2119, 20sylib 188 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( A  i^i  ( M `  (  ._|_  `  Q ) ) )  =  ( M `  (  ._|_  `  Q )
) )
227, 12, 213eqtrd 2332 1  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( M `  (  ._|_  `  Q ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   {csn 3653   |^|_ciin 3922   ` cfv 5271   Basecbs 13164   occoc 13232   OPcops 29984   OLcol 29986   Atomscatm 30075   pmapcpmap 30308   _|_ PcpolN 30713
This theorem is referenced by:  2polatN  30743
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
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-ral 2561  df-rex 2562  df-reu 2563  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-oposet 29988  df-ol 29990  df-ats 30079  df-pmap 30315  df-polarityN 30714
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