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Theorem psubclsetN 30418
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 2924 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 psubclset.c . . 3  |-  C  =  ( PSubCl `  K )
3 fveq2 5687 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 psubclset.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2454 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65sseq2d 3336 . . . . . 6  |-  ( k  =  K  ->  (
s  C_  ( Atoms `  k )  <->  s  C_  A ) )
7 fveq2 5687 . . . . . . . . 9  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ( _|_ P `  K ) )
8 psubclset.p . . . . . . . . 9  |-  ._|_  =  ( _|_ P `  K
)
97, 8syl6eqr 2454 . . . . . . . 8  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ._|_  )
109fveq1d 5689 . . . . . . . 8  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  s
)  =  (  ._|_  `  s ) )
119, 10fveq12d 5693 . . . . . . 7  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  ( 
._|_  `  (  ._|_  `  s
) ) )
1211eqeq1d 2412 . . . . . 6  |-  ( k  =  K  ->  (
( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s  <-> 
(  ._|_  `  (  ._|_  `  s ) )  =  s ) )
136, 12anbi12d 692 . . . . 5  |-  ( k  =  K  ->  (
( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) ) )
1413abbidv 2518 . . . 4  |-  ( k  =  K  ->  { s  |  ( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) } )
15 df-psubclN 30417 . . . 4  |-  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_ P `  k ) `
 ( ( _|_
P `  k ) `  s ) )  =  s ) } )
16 fvex 5701 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
174, 16eqeltri 2474 . . . . . 6  |-  A  e. 
_V
1817pwex 4342 . . . . 5  |-  ~P A  e.  _V
19 df-pw 3761 . . . . . . . . 9  |-  ~P A  =  { s  |  s 
C_  A }
2019abeq2i 2511 . . . . . . . 8  |-  ( s  e.  ~P A  <->  s  C_  A )
2120anbi1i 677 . . . . . . 7  |-  ( ( s  e.  ~P A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) )
2221abbii 2516 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }
23 ssab2 3387 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  C_  ~P A
2422, 23eqsstr3i 3339 . . . . 5  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } 
C_  ~P A
2518, 24ssexi 4308 . . . 4  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) }  e.  _V
2614, 15, 25fvmpt 5765 . . 3  |-  ( K  e.  _V  ->  ( PSubCl `
 K )  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
272, 26syl5eq 2448 . 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 359    = wceq 1649    e. wcel 1721   {cab 2390   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   ` cfv 5413   Atomscatm 29746   _|_ PcpolN 30384   PSubClcpscN 30416
This theorem is referenced by:  ispsubclN  30419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-psubclN 30417
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