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Theorem pointpsubN 29190
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 2258 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 pointpsub.p . . . 4  |-  P  =  ( Points `  K )
31, 2ispointN 29181 . . 3  |-  ( K  e.  AtLat  ->  ( X  e.  P  <->  E. q  e.  (
Atoms `  K ) X  =  { q } ) )
4 pointpsub.s . . . . . . 7  |-  S  =  ( PSubSp `  K )
51, 4snatpsubN 29189 . . . . . 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 2327 . . . . 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 2641 . . 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 2519   {csn 3614   ` cfv 4673   Atomscatm 28703   AtLatcal 28704   PointscpointsN 28934   PSubSpcpsubsp 28935
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-join 14073  df-lat 14115  df-covers 28706  df-ats 28707  df-atl 28738  df-pointsN 28941  df-psubsp 28942
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