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Theorem psubcli2N 36060
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubcli2.p  |-  ._|_  =  ( _|_P `  K
)
psubcli2.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
psubcli2N  |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )

Proof of Theorem psubcli2N
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 psubcli2.p . . 3  |-  ._|_  =  ( _|_P `  K
)
3 psubcli2.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 36058 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
54simplbda 622 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570   Atomscatm 35385   _|_PcpolN 36023   PSubClcpscN 36055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-psubclN 36056
This theorem is referenced by:  psubclsubN  36061  pmapidclN  36063  poml6N  36076  osumcllem3N  36079  osumclN  36088  pmapojoinN  36089  pexmidN  36090  pexmidlem6N  36096
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