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Theorem cssval 16582
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 2796 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 cssval.c . . 3  |-  C  =  ( CSubSp `  W )
3 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
4 cssval.o . . . . . . . 8  |-  ._|_  =  ( ocv `  W )
53, 4syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
65fveq1d 5527 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  s )  =  (  ._|_  `  s
) )
75, 6fveq12d 5531 . . . . . 6  |-  ( w  =  W  ->  (
( ocv `  w
) `  ( ( ocv `  w ) `  s ) )  =  (  ._|_  `  (  ._|_  `  s ) ) )
87eqeq2d 2294 . . . . 5  |-  ( w  =  W  ->  (
s  =  ( ( ocv `  w ) `
 ( ( ocv `  w ) `  s
) )  <->  s  =  (  ._|_  `  (  ._|_  `  s ) ) ) )
98abbidv 2397 . . . 4  |-  ( w  =  W  ->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) }  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
10 df-css 16564 . . . 4  |-  CSubSp  =  ( w  e.  _V  |->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) } )
11 fvex 5539 . . . . . 6  |-  ( Base `  W )  e.  _V
1211pwex 4193 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
13 id 19 . . . . . . 7  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  =  ( 
._|_  `  (  ._|_  `  s
) ) )
14 eqid 2283 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
1514, 4ocvss 16570 . . . . . . . 8  |-  (  ._|_  `  (  ._|_  `  s ) )  C_  ( Base `  W )
16 fvex 5539 . . . . . . . . 9  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  _V
1716elpw 3631 . . . . . . . 8  |-  ( ( 
._|_  `  (  ._|_  `  s
) )  e.  ~P ( Base `  W )  <->  ( 
._|_  `  (  ._|_  `  s
) )  C_  ( Base `  W ) )
1815, 17mpbir 200 . . . . . . 7  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  ~P ( Base `  W )
1913, 18syl6eqel 2371 . . . . . 6  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  e.  ~P ( Base `  W )
)
2019abssi 3248 . . . . 5  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  C_  ~P ( Base `  W
)
2112, 20ssexi 4159 . . . 4  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  e.  _V
229, 10, 21fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  ( CSubSp `
 W )  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
232, 22syl5eq 2327 . 2  |-  ( W  e.  _V  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
241, 23syl 15 1  |-  ( W  e.  X  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   ` cfv 5255   Basecbs 13148   ocvcocv 16560   CSubSpccss 16561
This theorem is referenced by:  iscss  16583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-ocv 16563  df-css 16564
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