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Theorem ispsubclN 30126
Description: The predicate "is 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
ispsubclN  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )

Proof of Theorem ispsubclN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 psubclset.a . . . 4  |-  A  =  ( Atoms `  K )
2 psubclset.p . . . 4  |-  ._|_  =  ( _|_ P `  K
)
3 psubclset.c . . . 4  |-  C  =  ( PSubCl `  K )
41, 2, 3psubclsetN 30125 . . 3  |-  ( K  e.  D  ->  C  =  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) } )
54eleq2d 2350 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  X  e.  { x  |  ( x 
C_  A  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) } ) )
6 fvex 5539 . . . . . 6  |-  ( Atoms `  K )  e.  _V
71, 6eqeltri 2353 . . . . 5  |-  A  e. 
_V
87ssex 4158 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
98adantr 451 . . 3  |-  ( ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  _V )
10 sseq1 3199 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 fveq2 5525 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  x )  =  (  ._|_  `  X ) )
1211fveq2d 5529 . . . . 5  |-  ( x  =  X  ->  (  ._|_  `  (  ._|_  `  x
) )  =  ( 
._|_  `  (  ._|_  `  X
) ) )
13 id 19 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1412, 13eqeq12d 2297 . . . 4  |-  ( x  =  X  ->  (
(  ._|_  `  (  ._|_  `  x ) )  =  x  <->  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
1510, 14anbi12d 691 . . 3  |-  ( x  =  X  ->  (
( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
169, 15elab3 2921 . 2  |-  ( X  e.  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) }  <-> 
( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
175, 16syl6bb 252 1  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152   ` cfv 5255   Atomscatm 29453   _|_
PcpolN 30091   PSubClcpscN 30123
This theorem is referenced by:  psubcliN  30127  psubcli2N  30128  0psubclN  30132  1psubclN  30133  atpsubclN  30134  pmapsubclN  30135  ispsubcl2N  30136  osumclN  30156  pexmidN  30158  pexmidlem6N  30164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-psubclN 30124
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