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Theorem linepsubclN 30748
Description: A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
linepsubcl.n  |-  N  =  ( Lines `  K )
linepsubcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
linepsubclN  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  e.  C )

Proof of Theorem linepsubclN
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 30161 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 eqid 2436 . . . . 5  |-  ( join `  K )  =  (
join `  K )
3 eqid 2436 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 linepsubcl.n . . . . 5  |-  N  =  ( Lines `  K )
5 eqid 2436 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
62, 3, 4, 5isline2 30571 . . . 4  |-  ( K  e.  Lat  ->  ( X  e.  N  <->  E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  X  =  ( ( pmap `  K ) `  ( p ( join `  K ) q ) ) ) ) )
71, 6syl 16 . . 3  |-  ( K  e.  HL  ->  ( X  e.  N  <->  E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  X  =  ( ( pmap `  K ) `  ( p ( join `  K ) q ) ) ) ) )
81adantr 452 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  K  e.  Lat )
9 eqid 2436 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 30087 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1110ad2antrl 709 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  p  e.  (
Base `  K )
)
129, 3atbase 30087 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  ( Base `  K )
)
1312ad2antll 710 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  q  e.  (
Base `  K )
)
149, 2latjcl 14479 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  p  e.  ( Base `  K )  /\  q  e.  ( Base `  K
) )  ->  (
p ( join `  K
) q )  e.  ( Base `  K
) )
158, 11, 13, 14syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( p (
join `  K )
q )  e.  (
Base `  K )
)
16 linepsubcl.c . . . . . . . 8  |-  C  =  ( PSubCl `  K )
179, 5, 16pmapsubclN 30743 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( p ( join `  K ) q )  e.  ( Base `  K
) )  ->  (
( pmap `  K ) `  ( p ( join `  K ) q ) )  e.  C )
1815, 17syldan 457 . . . . . 6  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( ( pmap `  K ) `  (
p ( join `  K
) q ) )  e.  C )
19 eleq1a 2505 . . . . . 6  |-  ( ( ( pmap `  K
) `  ( p
( join `  K )
q ) )  e.  C  ->  ( X  =  ( ( pmap `  K ) `  (
p ( join `  K
) q ) )  ->  X  e.  C
) )
2018, 19syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( X  =  ( ( pmap `  K
) `  ( p
( join `  K )
q ) )  ->  X  e.  C )
)
2120adantld 454 . . . 4  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( ( p  =/=  q  /\  X  =  ( ( pmap `  K ) `  (
p ( join `  K
) q ) ) )  ->  X  e.  C ) )
2221rexlimdvva 2837 . . 3  |-  ( K  e.  HL  ->  ( E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  X  =  ( ( pmap `  K ) `  (
p ( join `  K
) q ) ) )  ->  X  e.  C ) )
237, 22sylbid 207 . 2  |-  ( K  e.  HL  ->  ( X  e.  N  ->  X  e.  C ) )
2423imp 419 1  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   ` cfv 5454  (class class class)co 6081   Basecbs 13469   joincjn 14401   Latclat 14474   Atomscatm 30061   HLchlt 30148   Linesclines 30291   pmapcpmap 30294   PSubClcpscN 30731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lines 30298  df-pmap 30301  df-polarityN 30700  df-psubclN 30732
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