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Theorem pclssidN 30010
Description: A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
pclss.a  |-  A  =  ( Atoms `  K )
pclss.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
pclssidN  |-  ( ( K  e.  V  /\  X  C_  A )  ->  X  C_  ( U `  X ) )

Proof of Theorem pclssidN
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssintub 4011 . 2  |-  X  C_  |^|
{ y  e.  (
PSubSp `  K )  |  X  C_  y }
2 pclss.a . . 3  |-  A  =  ( Atoms `  K )
3 eqid 2388 . . 3  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
4 pclss.c . . 3  |-  U  =  ( PCl `  K
)
52, 3, 4pclvalN 30005 . 2  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
61, 5syl5sseqr 3341 1  |-  ( ( K  e.  V  /\  X  C_  A )  ->  X  C_  ( U `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654    C_ wss 3264   |^|cint 3993   ` cfv 5395   Atomscatm 29379   PSubSpcpsubsp 29611   PClcpclN 30002
This theorem is referenced by:  pclunN  30013  pcl0bN  30038  pclfinclN  30065
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-psubsp 29618  df-pclN 30003
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