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Theorem pol1N 30396
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 3331 . . 3  |-  A  C_  A
2 eqid 2408 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2408 . . . 4  |-  ( oc
`  K )  =  ( oc `  K
)
4 polssat.a . . . 4  |-  A  =  ( Atoms `  K )
5 eqid 2408 . . . 4  |-  ( pmap `  K )  =  (
pmap `  K )
6 polssat.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
72, 3, 4, 5, 6polval2N 30392 . . 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 29849 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
10 eqid 2408 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
1110, 4atbase 29776 . . . . . . . . . 10  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
12 eqid 2408 . . . . . . . . . . 11  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2408 . . . . . . . . . . 11  |-  ( 1.
`  K )  =  ( 1. `  K
)
1410, 12, 13ople1 29678 . . . . . . . . . 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 2753 . . . . . . . 8  |-  ( K  e.  HL  ->  A. p  e.  A  p ( le `  K ) ( 1. `  K ) )
17 rabid2 2849 . . . . . . . 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 5695 . . . . . 6  |-  ( K  e.  HL  ->  (
( lub `  K
) `  A )  =  ( ( lub `  K ) `  {
p  e.  A  |  p ( le `  K ) ( 1.
`  K ) } ) )
20 hlomcmat 29851 . . . . . . 7  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
) )
2110, 13op1cl 29672 . . . . . . . 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 29806 . . . . . . 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 2441 . . . . 5  |-  ( K  e.  HL  ->  ( 1. `  K )  =  ( ( lub `  K
) `  A )
)
2625fveq2d 5695 . . . 4  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( 1. `  K ) )  =  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )
27 eqid 2408 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2827, 13, 3opoc1 29689 . . . . 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 2442 . . 3  |-  ( K  e.  HL  ->  (
( oc `  K
) `  ( ( lub `  K ) `  A ) )  =  ( 0. `  K
) )
3130fveq2d 5695 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( ( oc `  K ) `  (
( lub `  K
) `  A )
) )  =  ( ( pmap `  K
) `  ( 0. `  K ) ) )
32 hlatl 29847 . . 3  |-  ( K  e.  HL  ->  K  e.  AtLat )
3327, 5pmap0 30251 . . 3  |-  ( K  e.  AtLat  ->  ( ( pmap `  K ) `  ( 0. `  K ) )  =  (/) )
3432, 33syl 16 . 2  |-  ( K  e.  HL  ->  (
( pmap `  K ) `  ( 0. `  K
) )  =  (/) )
358, 31, 343eqtrd 2444 1  |-  ( K  e.  HL  ->  (  ._|_  `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   {crab 2674    C_ wss 3284   (/)c0 3592   class class class wbr 4176   ` cfv 5417   Basecbs 13428   lecple 13495   occoc 13496   lubclub 14358   0.cp0 14425   1.cp1 14426   CLatccla 14495   OPcops 29659   OMLcoml 29662   Atomscatm 29750   AtLatcal 29751   HLchlt 29837   pmapcpmap 29983   _|_
PcpolN 30388
This theorem is referenced by:  2pol0N  30397  1psubclN  30430
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-pmap 29990  df-polarityN 30389
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