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Theorem linepsubclN 30762
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 30175 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
2 eqid 2296 . . . . 5  |-  ( join `  K )  =  (
join `  K )
3 eqid 2296 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 linepsubcl.n . . . . 5  |-  N  =  ( Lines `  K )
5 eqid 2296 . . . . 5  |-  ( pmap `  K )  =  (
pmap `  K )
62, 3, 4, 5isline2 30585 . . . 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 15 . . 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 451 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  K  e.  Lat )
9 eqid 2296 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 30101 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1110ad2antrl 708 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  p  e.  (
Base `  K )
)
129, 3atbase 30101 . . . . . . . . 9  |-  ( q  e.  ( Atoms `  K
)  ->  q  e.  ( Base `  K )
)
1312ad2antll 709 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  q  e.  (
Base `  K )
)
149, 2latjcl 14172 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  p  e.  ( Base `  K )  /\  q  e.  ( Base `  K
) )  ->  (
p ( join `  K
) q )  e.  ( Base `  K
) )
158, 11, 13, 14syl3anc 1182 . . . . . . 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 30757 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( p ( join `  K ) q )  e.  ( Base `  K
) )  ->  (
( pmap `  K ) `  ( p ( join `  K ) q ) )  e.  C )
1815, 17syldan 456 . . . . . 6  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( ( pmap `  K ) `  (
p ( join `  K
) q ) )  e.  C )
19 eleq1a 2365 . . . . . 6  |-  ( ( ( pmap `  K
) `  ( p
( join `  K )
q ) )  e.  C  ->  ( X  =  ( ( pmap `  K ) `  (
p ( join `  K
) q ) )  ->  X  e.  C
) )
2018, 19syl 15 . . . . 5  |-  ( ( K  e.  HL  /\  ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  ( X  =  ( ( pmap `  K
) `  ( p
( join `  K )
q ) )  ->  X  e.  C )
)
2120adantld 453 . . . 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 2687 . . 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 206 . 2  |-  ( K  e.  HL  ->  ( X  e.  N  ->  X  e.  C ) )
2423imp 418 1  |-  ( ( K  e.  HL  /\  X  e.  N )  ->  X  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   Linesclines 30305   pmapcpmap 30308   PSubClcpscN 30745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lines 30312  df-pmap 30315  df-polarityN 30714  df-psubclN 30746
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