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Theorem psubclsetN 30747
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 2809 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 psubclset.c . . 3  |-  C  =  ( PSubCl `  K )
3 fveq2 5541 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 psubclset.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2346 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65sseq2d 3219 . . . . . 6  |-  ( k  =  K  ->  (
s  C_  ( Atoms `  k )  <->  s  C_  A ) )
7 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ( _|_ P `  K ) )
8 psubclset.p . . . . . . . . 9  |-  ._|_  =  ( _|_ P `  K
)
97, 8syl6eqr 2346 . . . . . . . 8  |-  ( k  =  K  ->  ( _|_ P `  k )  =  ._|_  )
109fveq1d 5543 . . . . . . . 8  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  s
)  =  (  ._|_  `  s ) )
119, 10fveq12d 5547 . . . . . . 7  |-  ( k  =  K  ->  (
( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  ( 
._|_  `  (  ._|_  `  s
) ) )
1211eqeq1d 2304 . . . . . 6  |-  ( k  =  K  ->  (
( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s  <-> 
(  ._|_  `  (  ._|_  `  s ) )  =  s ) )
136, 12anbi12d 691 . . . . 5  |-  ( k  =  K  ->  (
( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) ) )
1413abbidv 2410 . . . 4  |-  ( k  =  K  ->  { s  |  ( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) } )
15 df-psubclN 30746 . . . 4  |-  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_ P `  k ) `
 ( ( _|_
P `  k ) `  s ) )  =  s ) } )
16 fvex 5555 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
174, 16eqeltri 2366 . . . . . 6  |-  A  e. 
_V
1817pwex 4209 . . . . 5  |-  ~P A  e.  _V
19 df-pw 3640 . . . . . . . . 9  |-  ~P A  =  { s  |  s 
C_  A }
2019abeq2i 2403 . . . . . . . 8  |-  ( s  e.  ~P A  <->  s  C_  A )
2120anbi1i 676 . . . . . . 7  |-  ( ( s  e.  ~P A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) )
2221abbii 2408 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }
23 ssab2 3270 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  C_  ~P A
2422, 23eqsstr3i 3222 . . . . 5  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } 
C_  ~P A
2518, 24ssexi 4175 . . . 4  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) }  e.  _V
2614, 15, 25fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( PSubCl `
 K )  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
272, 26syl5eq 2340 . 2  |-  ( K  e.  _V  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
281, 27syl 15 1  |-  ( K  e.  B  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   ` cfv 5271   Atomscatm 30075   _|_ PcpolN 30713   PSubClcpscN 30745
This theorem is referenced by:  ispsubclN  30748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-psubclN 30746
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