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Theorem psubssat 29943
Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atpsub.a  |-  A  =  ( Atoms `  K )
atpsub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
psubssat  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  A )

Proof of Theorem psubssat
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
3 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
4 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4ispsubsp 29934 . 2  |-  ( K  e.  B  ->  ( X  e.  S  <->  ( X  C_  A  /\  A. p  e.  X  A. q  e.  X  A. r  e.  A  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  X ) ) ) )
65simprbda 606 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   PSubSpcpsubsp 29685
This theorem is referenced by:  psubatN  29944  paddidm  30030  paddclN  30031  paddss  30034  pmodlem1  30035  pmod1i  30037  pmodl42N  30040  elpcliN  30082  pclidN  30085  pclbtwnN  30086  pclunN  30087  pclun2N  30088  pclfinN  30089  polssatN  30097  psubclsubN  30129
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-psubsp 29692
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