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Theorem psubssat 35930
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 2396 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2396 . . 3  |-  ( join `  K )  =  (
join `  K )
3 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
4 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4ispsubsp 35921 . 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 621 1  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746    C_ wss 3406   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   lecple 14732   joincjn 15713   Atomscatm 35440   PSubSpcpsubsp 35672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-iota 5477  df-fun 5515  df-fv 5521  df-ov 6221  df-psubsp 35679
This theorem is referenced by:  psubatN  35931  paddidm  36017  paddclN  36018  paddss  36021  pmodlem1  36022  pmod1i  36024  pmodl42N  36027  elpcliN  36069  pclidN  36072  pclbtwnN  36073  pclunN  36074  pclun2N  36075  pclfinN  36076  polssatN  36084  psubclsubN  36116
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