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Theorem psubclsetN 30734
Description: The set of closed projective subspaces in a Hilbert lattice. (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
psubclsetN  |-  ( K  e.  B  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
Distinct variable groups:    A, s    K, s
Allowed substitution hints:    B( s)    C( s)   
._|_ ( s)

Proof of Theorem psubclsetN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2965 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 psubclset.c . . 3  |-  C  =  ( PSubCl `  K )
3 fveq2 5729 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 psubclset.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2487 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65sseq2d 3377 . . . . . 6  |-  ( k  =  K  ->  (
s  C_  ( Atoms `  k )  <->  s  C_  A ) )
7 fveq2 5729 . . . . . . . . 9  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ( _|_ P `  K ) )
8 psubclset.p . . . . . . . . 9  |-  ._|_  =  ( _|_ P `  K
)
97, 8syl6eqr 2487 . . . . . . . 8  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ._|_  )
109fveq1d 5731 . . . . . . . 8  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  s
)  =  (  ._|_  `  s ) )
119, 10fveq12d 5735 . . . . . . 7  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  ( 
._|_  `  (  ._|_  `  s
) ) )
1211eqeq1d 2445 . . . . . 6  |-  ( k  =  K  ->  (
( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s  <-> 
(  ._|_  `  (  ._|_  `  s ) )  =  s ) )
136, 12anbi12d 693 . . . . 5  |-  ( k  =  K  ->  (
( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) ) )
1413abbidv 2551 . . . 4  |-  ( k  =  K  ->  { s  |  ( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) } )
15 df-psubclN 30733 . . . 4  |-  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_ P `  k ) `
 ( ( _|_
P `  k ) `  s ) )  =  s ) } )
16 fvex 5743 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
174, 16eqeltri 2507 . . . . . 6  |-  A  e. 
_V
1817pwex 4383 . . . . 5  |-  ~P A  e.  _V
19 df-pw 3802 . . . . . . . . 9  |-  ~P A  =  { s  |  s 
C_  A }
2019abeq2i 2544 . . . . . . . 8  |-  ( s  e.  ~P A  <->  s  C_  A )
2120anbi1i 678 . . . . . . 7  |-  ( ( s  e.  ~P A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) )
2221abbii 2549 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }
23 ssab2 3428 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  C_  ~P A
2422, 23eqsstr3i 3380 . . . . 5  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } 
C_  ~P A
2518, 24ssexi 4349 . . . 4  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) }  e.  _V
2614, 15, 25fvmpt 5807 . . 3  |-  ( K  e.  _V  ->  ( PSubCl `
 K )  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
272, 26syl5eq 2481 . 2  |-  ( K  e.  _V  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
281, 27syl 16 1  |-  ( K  e.  B  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2423   _Vcvv 2957    C_ wss 3321   ~Pcpw 3800   ` cfv 5455   Atomscatm 30062   _|_ PcpolN 30700   PSubClcpscN 30732
This theorem is referenced by:  ispsubclN  30735
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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-psubclN 30733
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