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Theorem pol1N 30075
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 3303 . . 3  |-  A  C_  A
2 eqid 2380 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2380 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
4 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
5 eqid 2380 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
6 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
72, 3, 4, 5, 6polval2N 30071 . . 3  |-  ( ( K  e.  HL  /\  A  C_  A )  -> 
(  ._|_  `  A )  =  ( ( pmap `  K ) `  (
( oc `  K
) `  ( ( lub `  K ) `  A ) ) ) )
81, 7mpan2 653 . 2  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  ( ( pmap `  K
) `  ( ( oc `  K ) `  ( ( lub `  K
) `  A )
) ) )
9 hlop 29528 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2380 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
1110, 4atbase 29455 . . . . . . . . . 10  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
12 eqid 2380 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2380 . . . . . . . . . . 11  |-  ( 1.
`  K )  =  ( 1. `  K
)
1410, 12, 13ople1 29357 . . . . . . . . . 10  |-  ( ( K  e.  OP  /\  p  e.  ( Base `  K ) )  ->  p ( le `  K ) ( 1.
`  K ) )
159, 11, 14syl2an 464 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  p  e.  A )  ->  p ( le `  K ) ( 1.
`  K ) )
1615ralrimiva 2725 . . . . . . . 8  |-  ( K  e.  HL  ->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
17 rabid2 2821 . . . . . . . 8  |-  ( A  =  { p  e.  A  |  p ( le `  K ) ( 1. `  K
) }  <->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
1816, 17sylibr 204 . . . . . . 7  |-  ( K  e.  HL  ->  A  =  { p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } )
1918fveq2d 5665 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  A )  =  ( ( lub `  K ) `  {
p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } ) )
20 hlomcmat 29530 . . . . . . 7  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2110, 13op1cl 29351 . . . . . . . 8  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
229, 21syl 16 . . . . . . 7  |-  ( K  e.  HL  ->  ( 1. `  K )  e.  ( Base `  K
) )
2310, 12, 2, 4atlatmstc 29485 . . . . . . 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 643 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  { p  e.  A  |  p
( le `  K
) ( 1. `  K ) } )  =  ( 1. `  K ) )
2519, 24eqtr2d 2413 . . . . 5  |-  ( K  e.  HL  ->  ( 1. `  K )  =  ( ( lub `  K
) `  A )
)
2625fveq2d 5665 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )
27 eqid 2380 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 13, 3opoc1 29368 . . . . 5  |-  ( K  e.  OP  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( 0. `  K
) )
299, 28syl 16 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( 0. `  K
) )
3026, 29eqtr3d 2414 . . 3  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( ( lub `  K ) `  A ) )  =  ( 0. `  K
) )
3130fveq2d 5665 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )  =  ( ( pmap `  K
) `  ( 0. `  K ) ) )
32 hlatl 29526 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
3327, 5pmap0 29930 . . 3  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 16 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( 0. `  K
) )  =  (/) )
358, 31, 343eqtrd 2416 1  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646    C_ wss 3256   (/)c0 3564   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   occoc 13457   lubclub 14319   0.cp0 14386   1.cp1 14387   CLatccla 14456   OPcops 29338   OMLcoml 29341   Atomscatm 29429   AtLatcal 29430   HLchlt 29516   pmapcpmap 29662   _|_
PcpolN 30067
This theorem is referenced by:  2pol0N  30076  1psubclN  30109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-pmap 29669  df-polarityN 30068
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