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Theorem cssval 16872
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 2932 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 cssval.c . . 3  |-  C  =  ( CSubSp `  W )
3 fveq2 5695 . . . . . . . 8  |-  ( w  =  W  ->  ( ocv `  w )  =  ( ocv `  W
) )
4 cssval.o . . . . . . . 8  |-  ._|_  =  ( ocv `  W )
53, 4syl6eqr 2462 . . . . . . 7  |-  ( w  =  W  ->  ( ocv `  w )  = 
._|_  )
65fveq1d 5697 . . . . . . 7  |-  ( w  =  W  ->  (
( ocv `  w
) `  s )  =  (  ._|_  `  s
) )
75, 6fveq12d 5701 . . . . . 6  |-  ( w  =  W  ->  (
( ocv `  w
) `  ( ( ocv `  w ) `  s ) )  =  (  ._|_  `  (  ._|_  `  s ) ) )
87eqeq2d 2423 . . . . 5  |-  ( w  =  W  ->  (
s  =  ( ( ocv `  w ) `
 ( ( ocv `  w ) `  s
) )  <->  s  =  (  ._|_  `  (  ._|_  `  s ) ) ) )
98abbidv 2526 . . . 4  |-  ( w  =  W  ->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) }  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
10 df-css 16854 . . . 4  |-  CSubSp  =  ( w  e.  _V  |->  { s  |  s  =  ( ( ocv `  w
) `  ( ( ocv `  w ) `  s ) ) } )
11 fvex 5709 . . . . . 6  |-  ( Base `  W )  e.  _V
1211pwex 4350 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
13 id 20 . . . . . . 7  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  =  ( 
._|_  `  (  ._|_  `  s
) ) )
14 eqid 2412 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
1514, 4ocvss 16860 . . . . . . . 8  |-  (  ._|_  `  (  ._|_  `  s ) )  C_  ( Base `  W )
16 fvex 5709 . . . . . . . . 9  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  _V
1716elpw 3773 . . . . . . . 8  |-  ( ( 
._|_  `  (  ._|_  `  s
) )  e.  ~P ( Base `  W )  <->  ( 
._|_  `  (  ._|_  `  s
) )  C_  ( Base `  W ) )
1815, 17mpbir 201 . . . . . . 7  |-  (  ._|_  `  (  ._|_  `  s ) )  e.  ~P ( Base `  W )
1913, 18syl6eqel 2500 . . . . . 6  |-  ( s  =  (  ._|_  `  (  ._|_  `  s ) )  ->  s  e.  ~P ( Base `  W )
)
2019abssi 3386 . . . . 5  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  C_  ~P ( Base `  W
)
2112, 20ssexi 4316 . . . 4  |-  { s  |  s  =  ( 
._|_  `  (  ._|_  `  s
) ) }  e.  _V
229, 10, 21fvmpt 5773 . . 3  |-  ( W  e.  _V  ->  ( CSubSp `
 W )  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
232, 22syl5eq 2456 . 2  |-  ( W  e.  _V  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
241, 23syl 16 1  |-  ( W  e.  X  ->  C  =  { s  |  s  =  (  ._|_  `  (  ._|_  `  s ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {cab 2398   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767   ` cfv 5421   Basecbs 13432   ocvcocv 16850   CSubSpccss 16851
This theorem is referenced by:  iscss  16873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-ocv 16853  df-css 16854
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