Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pol1N Unicode version

Theorem pol1N 30721
Description: The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polssat.a  |-  A  =  ( Atoms `  K )
polssat.p  |-  ._|_  =  ( _|_ P `  K
)
Assertion
Ref Expression
pol1N  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )

Proof of Theorem pol1N
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 ssid 3210 . . 3  |-  A  C_  A
2 eqid 2296 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2296 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
4 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
5 eqid 2296 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
6 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
72, 3, 4, 5, 6polval2N 30717 . . 3  |-  ( ( K  e.  HL  /\  A  C_  A )  -> 
(  ._|_  `  A )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  A ) ) ) )
81, 7mpan2 652 . 2  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  A )
) ) )
9 hlop 30174 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
1110, 4atbase 30101 . . . . . . . . . 10  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
12 eqid 2296 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2296 . . . . . . . . . . 11  |-  ( 1.
`  K )  =  ( 1. `  K
)
1410, 12, 13ople1 30003 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  p ( le `  K ) ( 1.
`  K ) )
159, 11, 14syl2an 463 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A )  ->  p ( le `  K ) ( 1.
`  K ) )
1615ralrimiva 2639 . . . . . . . 8  |-  ( K  e.  HL  ->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
17 rabid2 2730 . . . . . . . 8  |-  ( A  =  { p  e.  A  |  p ( le `  K ) ( 1. `  K
) }  <->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
1816, 17sylibr 203 . . . . . . 7  |-  ( K  e.  HL  ->  A  =  { p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } )
1918fveq2d 5545 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  A )  =  ( ( lub `  K ) `  {
p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } ) )
20 hlomcmat 30176 . . . . . . 7  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2110, 13op1cl 29997 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
229, 21syl 15 . . . . . . 7  |-  ( K  e.  HL  ->  ( 1. `  K )  e.  ( Base `  K
) )
2310, 12, 2, 4atlatmstc 30131 . . . . . . 7  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  ( 1. `  K )  e.  ( Base `  K
) )  ->  (
( lub `  K
) `  { p  e.  A  |  p
( le `  K
) ( 1. `  K ) } )  =  ( 1. `  K ) )
2420, 22, 23syl2anc 642 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  { p  e.  A  |  p
( le `  K
) ( 1. `  K ) } )  =  ( 1. `  K ) )
2519, 24eqtr2d 2329 . . . . 5  |-  ( K  e.  HL  ->  ( 1. `  K )  =  ( ( lub `  K
) `  A )
)
2625fveq2d 5545 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )
27 eqid 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 13, 3opoc1 30014 . . . . 5  |-  ( K  e.  OP  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( 0. `  K
) )
299, 28syl 15 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( 0. `  K
) )
3026, 29eqtr3d 2330 . . 3  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( ( lub `  K ) `  A ) )  =  ( 0. `  K
) )
3130fveq2d 5545 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )  =  ( ( pmap `  K
) `  ( 0. `  K ) ) )
32 hlatl 30172 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
3327, 5pmap0 30576 . . 3  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 15 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( 0. `  K
) )  =  (/) )
358, 31, 343eqtrd 2332 1  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   occoc 13232   lubclub 14092   0.cp0 14159   1.cp1 14160   CLatccla 14229   OPcops 29984   OMLcoml 29987   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   pmapcpmap 30308   _|_
PcpolN 30713
This theorem is referenced by:  2pol0N  30722  1psubclN  30755
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  ax-un 4528
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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-pmap 30315  df-polarityN 30714
  Copyright terms: Public domain W3C validator