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

Theorem polatN 30790
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 3944 . . 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 30764 . . 3  |-  ( ( K  e.  OL  /\  { Q }  C_  A
)  ->  ( P `  { Q } )  =  ( A  i^i  |^|_
p  e.  { Q }  ( M `  (  ._|_  `  p )
) ) )
71, 6sylan2 462 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( A  i^i  |^|_ p  e.  { Q }  ( M `  (  ._|_  `  p ) ) ) )
8 fveq2 5730 . . . . . 6  |-  ( p  =  Q  ->  (  ._|_  `  p )  =  (  ._|_  `  Q ) )
98fveq2d 5734 . . . . 5  |-  ( p  =  Q  ->  ( M `  (  ._|_  `  p ) )  =  ( M `  (  ._|_  `  Q ) ) )
109iinxsng 4169 . . . 4  |-  ( Q  e.  A  ->  |^|_ p  e.  { Q }  ( M `  (  ._|_  `  p ) )  =  ( M `  (  ._|_  `  Q ) ) )
1110adantl 454 . . 3  |-  ( ( K  e.  OL  /\  Q  e.  A )  -> 
|^|_ p  e.  { Q }  ( M `  (  ._|_  `  p )
)  =  ( M `
 (  ._|_  `  Q
) ) )
1211ineq2d 3544 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( A  i^i  |^|_ p  e.  { Q } 
( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  ( M `  (  ._|_  `  Q )
) ) )
13 olop 30074 . . . . 5  |-  ( K  e.  OL  ->  K  e.  OP )
14 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1514, 3atbase 30149 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 2opoccl 30054 . . . . 5  |-  ( ( K  e.  OP  /\  Q  e.  ( Base `  K ) )  -> 
(  ._|_  `  Q )  e.  ( Base `  K
) )
1713, 15, 16syl2an 465 . . . 4  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  (  ._|_  `  Q )  e.  ( Base `  K
) )
1814, 3, 4pmapssat 30618 . . . 4  |-  ( ( K  e.  OL  /\  (  ._|_  `  Q )  e.  ( Base `  K
) )  ->  ( M `  (  ._|_  `  Q ) )  C_  A )
1917, 18syldan 458 . . 3  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( M `  (  ._|_  `  Q ) ) 
C_  A )
20 sseqin2 3562 . . 3  |-  ( ( M `  (  ._|_  `  Q ) )  C_  A 
<->  ( A  i^i  ( M `  (  ._|_  `  Q ) ) )  =  ( M `  (  ._|_  `  Q )
) )
2119, 20sylib 190 . 2  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( A  i^i  ( M `  (  ._|_  `  Q ) ) )  =  ( M `  (  ._|_  `  Q )
) )
227, 12, 213eqtrd 2474 1  |-  ( ( K  e.  OL  /\  Q  e.  A )  ->  ( P `  { Q } )  =  ( M `  (  ._|_  `  Q ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   {csn 3816   |^|_ciin 4096   ` cfv 5456   Basecbs 13471   occoc 13539   OPcops 30032   OLcol 30034   Atomscatm 30123   pmapcpmap 30356   _|_ PcpolN 30761
This theorem is referenced by:  2polatN  30791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oposet 30036  df-ol 30038  df-ats 30127  df-pmap 30363  df-polarityN 30762
  Copyright terms: Public domain W3C validator