Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispsubclN Structured version   Unicode version

Theorem ispsubclN 36074
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 36073 . . 3  |-  ( K  e.  D  ->  C  =  { x  |  ( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) } )
54eleq2d 2452 . 2  |-  ( K  e.  D  ->  ( X  e.  C  <->  X  e.  { x  |  ( x 
C_  A  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) } ) )
6 fvex 5784 . . . . . 6  |-  ( Atoms `  K )  e.  _V
71, 6eqeltri 2466 . . . . 5  |-  A  e. 
_V
87ssex 4509 . . . 4  |-  ( X 
C_  A  ->  X  e.  _V )
98adantr 463 . . 3  |-  ( ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X )  ->  X  e.  _V )
10 sseq1 3438 . . . 4  |-  ( x  =  X  ->  (
x  C_  A  <->  X  C_  A
) )
11 fveq2 5774 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  x )  =  (  ._|_  `  X ) )
1211fveq2d 5778 . . . . 5  |-  ( x  =  X  ->  (  ._|_  `  (  ._|_  `  x
) )  =  ( 
._|_  `  (  ._|_  `  X
) ) )
13 id 22 . . . . 5  |-  ( x  =  X  ->  x  =  X )
1412, 13eqeq12d 2404 . . . 4  |-  ( x  =  X  ->  (
(  ._|_  `  (  ._|_  `  x ) )  =  x  <->  (  ._|_  `  (  ._|_  `  X ) )  =  X ) )
1510, 14anbi12d 708 . . 3  |-  ( x  =  X  ->  (
( x  C_  A  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x )  <->  ( X  C_  A  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X ) ) )
169, 15elab3 3178 . 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 367    = wceq 1399    e. wcel 1826   {cab 2367   _Vcvv 3034    C_ wss 3389   ` cfv 5496   Atomscatm 35401   _|_PcpolN 36039   PSubClcpscN 36071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-psubclN 36072
This theorem is referenced by:  psubcliN  36075  psubcli2N  36076  0psubclN  36080  1psubclN  36081  atpsubclN  36082  pmapsubclN  36083  ispsubcl2N  36084  osumclN  36104  pexmidN  36106  pexmidlem6N  36112
  Copyright terms: Public domain W3C validator