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Theorem psubssat 33737
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 2454 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2454 . . 3  |-  ( join `  K )  =  (
join `  K )
3 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
4 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4ispsubsp 33728 . 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 623 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3437   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   lecple 14365   joincjn 15234   Atomscatm 33247   PSubSpcpsubsp 33479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-psubsp 33486
This theorem is referenced by:  psubatN  33738  paddidm  33824  paddclN  33825  paddss  33828  pmodlem1  33829  pmod1i  33831  pmodl42N  33834  elpcliN  33876  pclidN  33879  pclbtwnN  33880  pclunN  33881  pclun2N  33882  pclfinN  33883  polssatN  33891  psubclsubN  33923
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