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Theorem cssval 18508
Description: The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssval.o  |-  ._|_  =  ( ocv `  W )
cssval.c  |-  C  =  ( CSubSp `  W )
Assertion
Ref Expression
cssval  |-  ( W  e.  X  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
Distinct variable groups:    ._|_ , s    W, s
Allowed substitution hints:    C( s)    X( s)

Proof of Theorem cssval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 3122 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 cssval.c . . 3  |-  C  =  ( CSubSp `  W )
3 fveq2 5866 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
4 cssval.o . . . . . . . 8  |-  ._|_  =  ( ocv `  W )
53, 4syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
65fveq1d 5868 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  s )  =  (  ._|_  `  s
) )
75, 6fveq12d 5872 . . . . . 6  |-  ( w  =  W  ->  (
( ocv `  w
) `  ( ( ocv `  w ) `  s ) )  =  (  ._|_  `  (  ._|_  `  s ) ) )
87eqeq2d 2481 . . . . 5  |-  ( w  =  W  ->  (
s  =  ( ( ocv `  w ) `
 ( ( ocv `  w ) `  s
) )  <->  s  =  (  ._|_  `  (  ._|_  `  s ) ) ) )
98abbidv 2603 . . . 4  |-  ( w  =  W  ->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) }  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
10 df-css 18490 . . . 4  |-  CSubSp  =  ( w  e.  _V  |->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) } )
11 fvex 5876 . . . . . 6  |-  ( Base `  W )  e.  _V
1211pwex 4630 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
13 id 22 . . . . . . 7  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  =  ( 
._|_  `  (  ._|_  `  s
) ) )
14 eqid 2467 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
1514, 4ocvss 18496 . . . . . . . 8  |-  (  ._|_  `  (  ._|_  `  s ) )  C_  ( Base `  W )
16 fvex 5876 . . . . . . . . 9  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  _V
1716elpw 4016 . . . . . . . 8  |-  ( ( 
._|_  `  (  ._|_  `  s
) )  e.  ~P ( Base `  W )  <->  ( 
._|_  `  (  ._|_  `  s
) )  C_  ( Base `  W ) )
1815, 17mpbir 209 . . . . . . 7  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  ~P ( Base `  W )
1913, 18syl6eqel 2563 . . . . . 6  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  e.  ~P ( Base `  W )
)
2019abssi 3575 . . . . 5  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  C_  ~P ( Base `  W
)
2112, 20ssexi 4592 . . . 4  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  e.  _V
229, 10, 21fvmpt 5950 . . 3  |-  ( W  e.  _V  ->  ( CSubSp `
 W )  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
232, 22syl5eq 2520 . 2  |-  ( W  e.  _V  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
241, 23syl 16 1  |-  ( W  e.  X  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   ` cfv 5588   Basecbs 14490   ocvcocv 18486   CSubSpccss 18487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-ocv 18489  df-css 18490
This theorem is referenced by:  iscss  18509
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