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Theorem psubcli2N 30425
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 2408 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 psubcli2.p . . 3  |-  ._|_  =  ( _|_ P `  K
)
3 psubcli2.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 30423 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
54simplbda 608 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3284   ` cfv 5417   Atomscatm 29750   _|_ PcpolN 30388   PSubClcpscN 30420
This theorem is referenced by:  psubclsubN  30426  pmapidclN  30428  poml6N  30441  osumcllem3N  30444  osumclN  30453  pmapojoinN  30454  pexmidN  30455  pexmidlem6N  30461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-psubclN 30421
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