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Theorem pointpsubN 29070
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
StepHypRef Expression
1 eqid 2256 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pointpsub.p . . . 4  |-  P  =  ( Points `  K )
31, 2ispointN 29061 . . 3  |-  ( K  e.  AtLat  ->  ( X  e.  P  <->  E. q  e.  (
Atoms `  K ) X  =  { q } ) )
4 pointpsub.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
51, 4snatpsubN 29069 . . . . . 6  |-  ( ( K  e.  AtLat  /\  q  e.  ( Atoms `  K )
)  ->  { q }  e.  S )
65ex 425 . . . . 5  |-  ( K  e.  AtLat  ->  ( q  e.  ( Atoms `  K )  ->  { q }  e.  S ) )
7 eleq1a 2325 . . . . 5  |-  ( { q }  e.  S  ->  ( X  =  {
q }  ->  X  e.  S ) )
86, 7syl6 31 . . . 4  |-  ( K  e.  AtLat  ->  ( q  e.  ( Atoms `  K )  ->  ( X  =  {
q }  ->  X  e.  S ) ) )
98rexlimdv 2637 . . 3  |-  ( K  e.  AtLat  ->  ( E. q  e.  ( Atoms `  K ) X  =  { q }  ->  X  e.  S ) )
103, 9sylbid 208 . 2  |-  ( K  e.  AtLat  ->  ( X  e.  P  ->  X  e.  S ) )
1110imp 420 1  |-  ( ( K  e.  AtLat  /\  X  e.  P )  ->  X  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517   {csn 3581   ` cfv 4638   Atomscatm 28583   AtLatcal 28584   PointscpointsN 28814   PSubSpcpsubsp 28815
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-join 14037  df-lat 14079  df-covers 28586  df-ats 28587  df-atl 28618  df-pointsN 28821  df-psubsp 28822
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