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Theorem ispsubclN 30796
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 30795 . . 3  |-  ( K  e.  D  ->  C  =  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) } )
54eleq2d 2505 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  X  e.  { x  |  ( x 
C_  A  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) } ) )
6 fvex 5744 . . . . . 6  |-  ( Atoms `  K )  e.  _V
71, 6eqeltri 2508 . . . . 5  |-  A  e. 
_V
87ssex 4349 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
98adantr 453 . . 3  |-  ( ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  _V )
10 sseq1 3371 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 fveq2 5730 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  x )  =  (  ._|_  `  X ) )
1211fveq2d 5734 . . . . 5  |-  ( x  =  X  ->  (  ._|_  `  (  ._|_  `  x
) )  =  ( 
._|_  `  (  ._|_  `  X
) ) )
13 id 21 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1412, 13eqeq12d 2452 . . . 4  |-  ( x  =  X  ->  (
(  ._|_  `  (  ._|_  `  x ) )  =  x  <->  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
1510, 14anbi12d 693 . . 3  |-  ( x  =  X  ->  (
( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
169, 15elab3 3091 . 2  |-  ( X  e.  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) }  <-> 
( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
175, 16syl6bb 254 1  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958    C_ wss 3322   ` cfv 5456   Atomscatm 30123   _|_
PcpolN 30761   PSubClcpscN 30793
This theorem is referenced by:  psubcliN  30797  psubcli2N  30798  0psubclN  30802  1psubclN  30803  atpsubclN  30804  pmapsubclN  30805  ispsubcl2N  30806  osumclN  30826  pexmidN  30828  pexmidlem6N  30834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-psubclN 30794
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