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Theorem psubclsubN 30055
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclsub.s  |-  S  =  ( PSubSp `  K )
psubclsub.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
psubclsubN  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  e.  S )

Proof of Theorem psubclsubN
StepHypRef Expression
1 eqid 2388 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
2 psubclsub.c . . 3  |-  C  =  ( PSubCl `  K )
31, 2psubcli2N 30054 . 2  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )
4 eqid 2388 . . . . . . 7  |-  ( Atoms `  K )  =  (
Atoms `  K )
54, 1, 2psubcliN 30053 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) )
65simpld 446 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  C_  ( Atoms `  K ) )
7 psubclsub.s . . . . . 6  |-  S  =  ( PSubSp `  K )
84, 7, 1polsubN 30022 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  e.  S )
96, 8syldan 457 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  X
)  e.  S )
104, 7psubssat 29869 . . . 4  |-  ( ( K  e.  HL  /\  ( ( _|_ P `  K ) `  X
)  e.  S )  ->  ( ( _|_
P `  K ) `  X )  C_  ( Atoms `  K ) )
119, 10syldan 457 . . 3  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )
124, 7, 1polsubN 30022 . . 3  |-  ( ( K  e.  HL  /\  ( ( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  e.  S
)
1311, 12syldan 457 . 2  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  e.  S
)
143, 13eqeltrrd 2463 1  |-  ( ( K  e.  HL  /\  X  e.  C )  ->  X  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264   ` cfv 5395   Atomscatm 29379   HLchlt 29466   PSubSpcpsubsp 29611   _|_ PcpolN 30017   PSubClcpscN 30049
This theorem is referenced by:  pclfinclN  30065
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-psubsp 29618  df-pmap 29619  df-polarityN 30018  df-psubclN 30050
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