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Theorem psubcli2N 30810
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 2438 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 psubcli2.p . . 3  |-  ._|_  =  ( _|_ P `  K
)
3 psubcli2.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 30808 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
54simplbda 609 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   ` cfv 5457   Atomscatm 30135   _|_ PcpolN 30773   PSubClcpscN 30805
This theorem is referenced by:  psubclsubN  30811  pmapidclN  30813  poml6N  30826  osumcllem3N  30829  osumclN  30838  pmapojoinN  30839  pexmidN  30840  pexmidlem6N  30846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-psubclN 30806
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