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Theorem ispsubclN 33863
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 33862 . . 3  |-  ( K  e.  D  ->  C  =  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) } )
54eleq2d 2519 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  X  e.  { x  |  ( x 
C_  A  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) } ) )
6 fvex 5785 . . . . . 6  |-  ( Atoms `  K )  e.  _V
71, 6eqeltri 2532 . . . . 5  |-  A  e. 
_V
87ssex 4520 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
98adantr 465 . . 3  |-  ( ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  _V )
10 sseq1 3461 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 fveq2 5775 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  x )  =  (  ._|_  `  X ) )
1211fveq2d 5779 . . . . 5  |-  ( x  =  X  ->  (  ._|_  `  (  ._|_  `  x
) )  =  ( 
._|_  `  (  ._|_  `  X
) ) )
13 id 22 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1412, 13eqeq12d 2471 . . . 4  |-  ( x  =  X  ->  (
(  ._|_  `  (  ._|_  `  x ) )  =  x  <->  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
1510, 14anbi12d 710 . . 3  |-  ( x  =  X  ->  (
( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
169, 15elab3 3196 . 2  |-  ( X  e.  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) }  <-> 
( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
175, 16syl6bb 261 1  |-  ( K  e.  D  ->  ( X  e.  C  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757   {cab 2435   _Vcvv 3054    C_ wss 3412   ` cfv 5502   Atomscatm 33190   _|_PcpolN 33828   PSubClcpscN 33860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-iota 5465  df-fun 5504  df-fv 5510  df-psubclN 33861
This theorem is referenced by:  psubcliN  33864  psubcli2N  33865  0psubclN  33869  1psubclN  33870  atpsubclN  33871  pmapsubclN  33872  ispsubcl2N  33873  osumclN  33893  pexmidN  33895  pexmidlem6N  33901
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