Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psubssat Unicode version

Theorem psubssat 29919
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 2380 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2380 . . 3  |-  ( join `  K )  =  (
join `  K )
3 atpsub.a . . 3  |-  A  =  ( Atoms `  K )
4 atpsub.s . . 3  |-  S  =  ( PSubSp `  K )
51, 2, 3, 4ispsubsp 29910 . 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 1649    e. wcel 1717   A.wral 2642    C_ wss 3256   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   lecple 13456   joincjn 14321   Atomscatm 29429   PSubSpcpsubsp 29661
This theorem is referenced by:  psubatN  29920  paddidm  30006  paddclN  30007  paddss  30010  pmodlem1  30011  pmod1i  30013  pmodl42N  30016  elpcliN  30058  pclidN  30061  pclbtwnN  30062  pclunN  30063  pclun2N  30064  pclfinN  30065  polssatN  30073  psubclsubN  30105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-psubsp 29668
  Copyright terms: Public domain W3C validator