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Theorem pol1N 29266
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 3172 . . 3  |-  A  C_  A
2 eqid 2258 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2258 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
4 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
5 eqid 2258 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
6 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
72, 3, 4, 5, 6polval2N 29262 . . 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 28719 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2258 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
1110, 4atbase 28646 . . . . . . . . . 10  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
12 eqid 2258 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2258 . . . . . . . . . . 11  |-  ( 1.
`  K )  =  ( 1. `  K
)
1410, 12, 13ople1 28548 . . . . . . . . . 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 2601 . . . . . . . 8  |-  ( K  e.  HL  ->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
17 rabid2 2692 . . . . . . . 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 5462 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  A )  =  ( ( lub `  K ) `  {
p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } ) )
20 hlomcmat 28721 . . . . . . 7  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2110, 13op1cl 28542 . . . . . . . 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 28676 . . . . . . 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 2291 . . . . 5  |-  ( K  e.  HL  ->  ( 1. `  K )  =  ( ( lub `  K
) `  A )
)
2625fveq2d 5462 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )
27 eqid 2258 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 13, 3opoc1 28559 . . . . 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 2292 . . 3  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( ( lub `  K ) `  A ) )  =  ( 0. `  K
) )
3130fveq2d 5462 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )  =  ( ( pmap `  K
) `  ( 0. `  K ) ) )
32 hlatl 28717 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
3327, 5pmap0 29121 . . 3  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 17 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( 0. `  K
) )  =  (/) )
358, 31, 343eqtrd 2294 1  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2518   {crab 2522    C_ wss 3127   (/)c0 3430   class class class wbr 3997   ` cfv 4673   Basecbs 13110   lecple 13177   occoc 13178   lubclub 14038   0.cp0 14105   1.cp1 14106   CLatccla 14175   OPcops 28529   OMLcoml 28532   Atomscatm 28620   AtLatcal 28621   HLchlt 28707   pmapcpmap 28853   _|_
PcpolN 29258
This theorem is referenced by:  2pol0N  29267  1psubclN  29300
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-pmap 28860  df-polarityN 29259
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