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Theorem pointpsubN 29219
Description: A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
pointpsub.p  |-  P  =  ( Points `  K )
pointpsub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
pointpsubN  |-  ( ( K  e.  AtLat  /\  X  e.  P )  ->  X  e.  S )

Proof of Theorem pointpsubN
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 eqid 2284 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pointpsub.p . . . 4  |-  P  =  ( Points `  K )
31, 2ispointN 29210 . . 3  |-  ( K  e.  AtLat  ->  ( X  e.  P  <->  E. q  e.  (
Atoms `  K ) X  =  { q } ) )
4 pointpsub.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
51, 4snatpsubN 29218 . . . . . 6  |-  ( ( K  e.  AtLat  /\  q  e.  ( Atoms `  K )
)  ->  { q }  e.  S )
65ex 423 . . . . 5  |-  ( K  e.  AtLat  ->  ( q  e.  ( Atoms `  K )  ->  { q }  e.  S ) )
7 eleq1a 2353 . . . . 5  |-  ( { q }  e.  S  ->  ( X  =  {
q }  ->  X  e.  S ) )
86, 7syl6 29 . . . 4  |-  ( K  e.  AtLat  ->  ( q  e.  ( Atoms `  K )  ->  ( X  =  {
q }  ->  X  e.  S ) ) )
98rexlimdv 2667 . . 3  |-  ( K  e.  AtLat  ->  ( E. q  e.  ( Atoms `  K ) X  =  { q }  ->  X  e.  S ) )
103, 9sylbid 206 . 2  |-  ( K  e.  AtLat  ->  ( X  e.  P  ->  X  e.  S ) )
1110imp 418 1  |-  ( ( K  e.  AtLat  /\  X  e.  P )  ->  X  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   E.wrex 2545   {csn 3641   ` cfv 5221   Atomscatm 28732   AtLatcal 28733   PointscpointsN 28963   PSubSpcpsubsp 28964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-join 14106  df-lat 14148  df-covers 28735  df-ats 28736  df-atl 28767  df-pointsN 28970  df-psubsp 28971
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