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Theorem psubcli2N 30197
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 2358 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 psubcli2.p . . 3  |-  ._|_  =  ( _|_ P `  K
)
3 psubcli2.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 30195 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
54simplbda 607 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228   ` cfv 5337   Atomscatm 29522   _|_ PcpolN 30160   PSubClcpscN 30192
This theorem is referenced by:  psubclsubN  30198  pmapidclN  30200  poml6N  30213  osumcllem3N  30216  osumclN  30225  pmapojoinN  30226  pexmidN  30227  pexmidlem6N  30233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-psubclN 30193
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