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Theorem ocvval 16567
Description: Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
ocvfval.v  |-  V  =  ( Base `  W
)
ocvfval.i  |-  .,  =  ( .i `  W )
ocvfval.f  |-  F  =  (Scalar `  W )
ocvfval.z  |-  .0.  =  ( 0g `  F )
ocvfval.o  |-  ._|_  =  ( ocv `  W )
Assertion
Ref Expression
ocvval  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
Distinct variable groups:    x, y,  .0.    x, V, y    x, W, y    x,  ., , y    x, S, y
Allowed substitution hints:    F( x, y)    ._|_ ( x, y)

Proof of Theorem ocvval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ocvfval.v . . . 4  |-  V  =  ( Base `  W
)
2 fvex 5539 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2353 . . 3  |-  V  e. 
_V
43elpw2 4175 . 2  |-  ( S  e.  ~P V  <->  S  C_  V
)
5 ocvfval.i . . . . . 6  |-  .,  =  ( .i `  W )
6 ocvfval.f . . . . . 6  |-  F  =  (Scalar `  W )
7 ocvfval.z . . . . . 6  |-  .0.  =  ( 0g `  F )
8 ocvfval.o . . . . . 6  |-  ._|_  =  ( ocv `  W )
91, 5, 6, 7, 8ocvfval 16566 . . . . 5  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
109fveq1d 5527 . . . 4  |-  ( W  e.  _V  ->  (  ._|_  `  S )  =  ( ( s  e. 
~P V  |->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  } ) `  S ) )
11 raleq 2736 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .,  y
)  =  .0.  <->  A. y  e.  S  ( x  .,  y )  =  .0.  ) )
1211rabbidv 2780 . . . . 5  |-  ( s  =  S  ->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  }  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
13 eqid 2283 . . . . 5  |-  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }
)  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }
)
143rabex 4165 . . . . 5  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  e.  _V
1512, 13, 14fvmpt 5602 . . . 4  |-  ( S  e.  ~P V  -> 
( ( s  e. 
~P V  |->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  } ) `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
1610, 15sylan9eq 2335 . . 3  |-  ( ( W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
17 fv01 5559 . . . . 5  |-  ( (/) `  S )  =  (/)
18 fvprc 5519 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
198, 18syl5eq 2327 . . . . . 6  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
2019fveq1d 5527 . . . . 5  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
21 ssrab2 3258 . . . . . 6  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  C_  V
22 fvprc 5519 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
231, 22syl5eq 2327 . . . . . 6  |-  ( -.  W  e.  _V  ->  V  =  (/) )
24 sseq0 3486 . . . . . 6  |-  ( ( { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  }  C_  V  /\  V  =  (/) )  ->  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0. 
}  =  (/) )
2521, 23, 24sylancr 644 . . . . 5  |-  ( -.  W  e.  _V  ->  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  }  =  (/) )
2617, 20, 253eqtr4a 2341 . . . 4  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2726adantr 451 . . 3  |-  ( ( -.  W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2816, 27pm2.61ian 765 . 2  |-  ( S  e.  ~P V  -> 
(  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
294, 28sylbir 204 1  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   ocvcocv 16560
This theorem is referenced by:  elocv  16568  ocv0  16577  csscld  18676  hlhilocv  31523
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
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