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Theorem psubclsetN 33546
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 3066 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 psubclset.c . . 3  |-  C  =  ( PSubCl `  K )
3 fveq2 5888 . . . . . . . 8  |-  ( k  =  K  ->  ( Atoms `  k )  =  ( Atoms `  K )
)
4 psubclset.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2514 . . . . . . 7  |-  ( k  =  K  ->  ( Atoms `  k )  =  A )
65sseq2d 3472 . . . . . 6  |-  ( k  =  K  ->  (
s  C_  ( Atoms `  k )  <->  s  C_  A ) )
7 fveq2 5888 . . . . . . . . 9  |-  ( k  =  K  ->  ( _|_P `  k )  =  ( _|_P `  K ) )
8 psubclset.p . . . . . . . . 9  |-  ._|_  =  ( _|_P `  K
)
97, 8syl6eqr 2514 . . . . . . . 8  |-  ( k  =  K  ->  ( _|_P `  k )  =  ._|_  )
109fveq1d 5890 . . . . . . . 8  |-  ( k  =  K  ->  (
( _|_P `  k ) `  s
)  =  (  ._|_  `  s ) )
119, 10fveq12d 5894 . . . . . . 7  |-  ( k  =  K  ->  (
( _|_P `  k ) `  (
( _|_P `  k ) `  s
) )  =  ( 
._|_  `  (  ._|_  `  s
) ) )
1211eqeq1d 2464 . . . . . 6  |-  ( k  =  K  ->  (
( ( _|_P `  k ) `  (
( _|_P `  k ) `  s
) )  =  s  <-> 
(  ._|_  `  (  ._|_  `  s ) )  =  s ) )
136, 12anbi12d 722 . . . . 5  |-  ( k  =  K  ->  (
( s  C_  ( Atoms `  k )  /\  ( ( _|_P `  k ) `  (
( _|_P `  k ) `  s
) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) ) )
1413abbidv 2580 . . . 4  |-  ( k  =  K  ->  { s  |  ( s  C_  ( Atoms `  k )  /\  ( ( _|_P `  k ) `  (
( _|_P `  k ) `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) } )
15 df-psubclN 33545 . . . 4  |-  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_P `  k ) `
 ( ( _|_P `  k ) `
 s ) )  =  s ) } )
16 fvex 5898 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
174, 16eqeltri 2536 . . . . . 6  |-  A  e. 
_V
1817pwex 4600 . . . . 5  |-  ~P A  e.  _V
19 selpw 3970 . . . . . . . 8  |-  ( s  e.  ~P A  <->  s  C_  A )
2019anbi1i 706 . . . . . . 7  |-  ( ( s  e.  ~P A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s )  <->  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) )
2120abbii 2578 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  =  {
s  |  ( s 
C_  A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }
22 ssab2 3525 . . . . . 6  |-  { s  |  ( s  e. 
~P A  /\  (  ._|_  `  (  ._|_  `  s
) )  =  s ) }  C_  ~P A
2321, 22eqsstr3i 3475 . . . . 5  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } 
C_  ~P A
2418, 23ssexi 4562 . . . 4  |-  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) }  e.  _V
2514, 15, 24fvmpt 5971 . . 3  |-  ( K  e.  _V  ->  ( PSubCl `
 K )  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
262, 25syl5eq 2508 . 2  |-  ( K  e.  _V  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
271, 26syl 17 1  |-  ( K  e.  B  ->  C  =  { s  |  ( s  C_  A  /\  (  ._|_  `  (  ._|_  `  s ) )  =  s ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   {cab 2448   _Vcvv 3057    C_ wss 3416   ~Pcpw 3963   ` cfv 5601   Atomscatm 32874   _|_PcpolN 33512   PSubClcpscN 33544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-psubclN 33545
This theorem is referenced by:  ispsubclN  33547
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