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Theorem pointpsubN 30013
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 2285 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pointpsub.p . . . 4  |-  P  =  ( Points `  K )
31, 2ispointN 30004 . . 3  |-  ( K  e.  AtLat  ->  ( X  e.  P  <->  E. q  e.  (
Atoms `  K ) X  =  { q } ) )
4 pointpsub.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
51, 4snatpsubN 30012 . . . . . 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 2354 . . . . 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 2668 . . 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 1625    e. wcel 1686   E.wrex 2546   {csn 3642   ` cfv 5257   Atomscatm 29526   AtLatcal 29527   PointscpointsN 29757   PSubSpcpsubsp 29758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-join 14112  df-lat 14154  df-covers 29529  df-ats 29530  df-atl 29561  df-pointsN 29764  df-psubsp 29765
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