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Theorem psubssat 30488
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 2435 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2435 . . 3  |-  ( join `  K )  =  (
join `  K )
3 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
4 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4ispsubsp 30479 . 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 607 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   Atomscatm 29998   PSubSpcpsubsp 30230
This theorem is referenced by:  psubatN  30489  paddidm  30575  paddclN  30576  paddss  30579  pmodlem1  30580  pmod1i  30582  pmodl42N  30585  elpcliN  30627  pclidN  30630  pclbtwnN  30631  pclunN  30632  pclun2N  30633  pclfinN  30634  polssatN  30642  psubclsubN  30674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-psubsp 30237
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