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Theorem cssval 16799
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 2881 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 cssval.c . . 3  |-  C  =  ( CSubSp `  W )
3 fveq2 5632 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
4 cssval.o . . . . . . . 8  |-  ._|_  =  ( ocv `  W )
53, 4syl6eqr 2416 . . . . . . 7  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
65fveq1d 5634 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  s )  =  (  ._|_  `  s
) )
75, 6fveq12d 5638 . . . . . 6  |-  ( w  =  W  ->  (
( ocv `  w
) `  ( ( ocv `  w ) `  s ) )  =  (  ._|_  `  (  ._|_  `  s ) ) )
87eqeq2d 2377 . . . . 5  |-  ( w  =  W  ->  (
s  =  ( ( ocv `  w ) `
 ( ( ocv `  w ) `  s
) )  <->  s  =  (  ._|_  `  (  ._|_  `  s ) ) ) )
98abbidv 2480 . . . 4  |-  ( w  =  W  ->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) }  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
10 df-css 16781 . . . 4  |-  CSubSp  =  ( w  e.  _V  |->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) } )
11 fvex 5646 . . . . . 6  |-  ( Base `  W )  e.  _V
1211pwex 4295 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
13 id 19 . . . . . . 7  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  =  ( 
._|_  `  (  ._|_  `  s
) ) )
14 eqid 2366 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
1514, 4ocvss 16787 . . . . . . . 8  |-  (  ._|_  `  (  ._|_  `  s ) )  C_  ( Base `  W )
16 fvex 5646 . . . . . . . . 9  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  _V
1716elpw 3720 . . . . . . . 8  |-  ( ( 
._|_  `  (  ._|_  `  s
) )  e.  ~P ( Base `  W )  <->  ( 
._|_  `  (  ._|_  `  s
) )  C_  ( Base `  W ) )
1815, 17mpbir 200 . . . . . . 7  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  ~P ( Base `  W )
1913, 18syl6eqel 2454 . . . . . 6  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  e.  ~P ( Base `  W )
)
2019abssi 3334 . . . . 5  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  C_  ~P ( Base `  W
)
2112, 20ssexi 4261 . . . 4  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  e.  _V
229, 10, 21fvmpt 5709 . . 3  |-  ( W  e.  _V  ->  ( CSubSp `
 W )  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
232, 22syl5eq 2410 . 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 1647    e. wcel 1715   {cab 2352   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   ` cfv 5358   Basecbs 13356   ocvcocv 16777   CSubSpccss 16778
This theorem is referenced by:  iscss  16800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-ocv 16780  df-css 16781
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