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Theorem polatN 30120
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 3759 . . 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 30094 . . 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 5525 . . . . . 6  |-  ( p  =  Q  ->  (  ._|_  `  p )  =  (  ._|_  `  Q ) )
98fveq2d 5529 . . . . 5  |-  ( p  =  Q  ->  ( M `  (  ._|_  `  p ) )  =  ( M `  (  ._|_  `  Q ) ) )
109iinxsng 3978 . . . 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 3370 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( A  i^i  |^|_ p  e.  { Q } 
( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  ( M `  (  ._|_  `  Q )
) ) )
13 olop 29404 . . . . 5  |-  ( K  e.  OL  ->  K  e.  OP )
14 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1514, 3atbase 29479 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 2opoccl 29384 . . . . 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 29948 . . . 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 3388 . . 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 2319 1  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( M `  (  ._|_  `  Q ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   {csn 3640   |^|_ciin 3906   ` cfv 5255   Basecbs 13148   occoc 13216   OPcops 29362   OLcol 29364   Atomscatm 29453   pmapcpmap 29686   _|_ PcpolN 30091
This theorem is referenced by:  2polatN  30121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oposet 29366  df-ol 29368  df-ats 29457  df-pmap 29693  df-polarityN 30092
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