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Theorem pointpsubN 28629
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 2253 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pointpsub.p . . . 4  |-  P  =  ( Points `  K )
31, 2ispointN 28620 . . 3  |-  ( K  e.  AtLat  ->  ( X  e.  P  <->  E. q  e.  (
Atoms `  K ) X  =  { q } ) )
4 pointpsub.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
51, 4snatpsubN 28628 . . . . . 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 2322 . . . . 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 2628 . . 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 2510   {csn 3544   ` cfv 4592   Atomscatm 28142   AtLatcal 28143   PointscpointsN 28373   PSubSpcpsubsp 28374
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-join 13954  df-lat 13996  df-covers 28145  df-ats 28146  df-atl 28177  df-pointsN 28380  df-psubsp 28381
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