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Theorem psubcliN 30735
Description: Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclset.a  |-  A  =  ( Atoms `  K )
psubclset.p  |-  ._|_  =  ( _|_ P `  K
)
psubclset.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
psubcliN  |-  ( ( K  e.  D  /\  X  e.  C )  ->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )

Proof of Theorem psubcliN
StepHypRef Expression
1 psubclset.a . . 3  |-  A  =  ( Atoms `  K )
2 psubclset.p . . 3  |-  ._|_  =  ( _|_ P `  K
)
3 psubclset.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 30734 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
54biimpa 471 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454   Atomscatm 30061   _|_ PcpolN 30699   PSubClcpscN 30731
This theorem is referenced by:  psubclsubN  30737  psubclssatN  30738
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-psubclN 30732
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