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Theorem pclssidN 30629
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 4060 . 2  |-  X  C_  |^|
{ y  e.  (
PSubSp `  K )  |  X  C_  y }
2 pclss.a . . 3  |-  A  =  ( Atoms `  K )
3 eqid 2435 . . 3  |-  ( PSubSp `  K )  =  (
PSubSp `  K )
4 pclss.c . . 3  |-  U  =  ( PCl `  K
)
52, 3, 4pclvalN 30624 . 2  |-  ( ( K  e.  V  /\  X  C_  A )  -> 
( U `  X
)  =  |^| { y  e.  ( PSubSp `  K
)  |  X  C_  y } )
61, 5syl5sseqr 3389 1  |-  ( ( K  e.  V  /\  X  C_  A )  ->  X  C_  ( U `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   |^|cint 4042   ` cfv 5446   Atomscatm 29998   PSubSpcpsubsp 30230   PClcpclN 30621
This theorem is referenced by:  pclunN  30632  pcl0bN  30657  pclfinclN  30684
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-int 4043  df-iun 4087  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-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-psubsp 30237  df-pclN 30622
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