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Theorem psubcli2N 33902
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 2452 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 psubcli2.p . . 3  |-  ._|_  =  ( _|_P `  K
)
3 psubcli2.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 33900 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
54simplbda 624 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3431   ` cfv 5521   Atomscatm 33227   _|_PcpolN 33865   PSubClcpscN 33897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-psubclN 33898
This theorem is referenced by:  psubclsubN  33903  pmapidclN  33905  poml6N  33918  osumcllem3N  33921  osumclN  33930  pmapojoinN  33931  pexmidN  33932  pexmidlem6N  33938
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