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Theorem pol1N 29366
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
StepHypRef Expression
1 ssid 3198 . . 3  |-  A  C_  A
2 eqid 2284 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2284 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
4 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
5 eqid 2284 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
6 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
72, 3, 4, 5, 6polval2N 29362 . . 3  |-  ( ( K  e.  HL  /\  A  C_  A )  -> 
(  ._|_  `  A )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  A ) ) ) )
81, 7mpan2 655 . 2  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  A )
) ) )
9 hlop 28819 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2284 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
1110, 4atbase 28746 . . . . . . . . . 10  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
12 eqid 2284 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2284 . . . . . . . . . . 11  |-  ( 1.
`  K )  =  ( 1. `  K
)
1410, 12, 13ople1 28648 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  p ( le `  K ) ( 1.
`  K ) )
159, 11, 14syl2an 465 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A )  ->  p ( le `  K ) ( 1.
`  K ) )
1615ralrimiva 2627 . . . . . . . 8  |-  ( K  e.  HL  ->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
17 rabid2 2718 . . . . . . . 8  |-  ( A  =  { p  e.  A  |  p ( le `  K ) ( 1. `  K
) }  <->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
1816, 17sylibr 205 . . . . . . 7  |-  ( K  e.  HL  ->  A  =  { p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } )
1918fveq2d 5489 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  A )  =  ( ( lub `  K ) `  {
p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } ) )
20 hlomcmat 28821 . . . . . . 7  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2110, 13op1cl 28642 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
229, 21syl 17 . . . . . . 7  |-  ( K  e.  HL  ->  ( 1. `  K )  e.  ( Base `  K
) )
2310, 12, 2, 4atlatmstc 28776 . . . . . . 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 645 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  { p  e.  A  |  p
( le `  K
) ( 1. `  K ) } )  =  ( 1. `  K ) )
2519, 24eqtr2d 2317 . . . . 5  |-  ( K  e.  HL  ->  ( 1. `  K )  =  ( ( lub `  K
) `  A )
)
2625fveq2d 5489 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )
27 eqid 2284 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 13, 3opoc1 28659 . . . . 5  |-  ( K  e.  OP  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( 0. `  K
) )
299, 28syl 17 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( 0. `  K
) )
3026, 29eqtr3d 2318 . . 3  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( ( lub `  K ) `  A ) )  =  ( 0. `  K
) )
3130fveq2d 5489 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )  =  ( ( pmap `  K
) `  ( 0. `  K ) ) )
32 hlatl 28817 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
3327, 5pmap0 29221 . . 3  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 17 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( 0. `  K
) )  =  (/) )
358, 31, 343eqtrd 2320 1  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1628    e. wcel 1688   A.wral 2544   {crab 2548    C_ wss 3153   (/)c0 3456   class class class wbr 4024   ` cfv 5221   Basecbs 13142   lecple 13209   occoc 13210   lubclub 14070   0.cp0 14137   1.cp1 14138   CLatccla 14207   OPcops 28629   OMLcoml 28632   Atomscatm 28720   AtLatcal 28721   HLchlt 28807   pmapcpmap 28953   _|_
PcpolN 29358
This theorem is referenced by:  2pol0N  29367  1psubclN  29400
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-pmap 28960  df-polarityN 29359
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