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Theorem ocvval 19161
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 5891 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2513 . . 3  |-  V  e. 
_V
43elpw2 4589 . 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 19160 . . . . 5  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
109fveq1d 5883 . . . 4  |-  ( W  e.  _V  ->  (  ._|_  `  S )  =  ( ( s  e. 
~P V  |->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  } ) `  S ) )
11 raleq 3032 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .,  y
)  =  .0.  <->  A. y  e.  S  ( x  .,  y )  =  .0.  ) )
1211rabbidv 3079 . . . . 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 2429 . . . . 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 4576 . . . . 5  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  e.  _V
1512, 13, 14fvmpt 5964 . . . 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 2490 . . 3  |-  ( ( W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
17 0fv 5914 . . . . 5  |-  ( (/) `  S )  =  (/)
18 fvprc 5875 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
198, 18syl5eq 2482 . . . . . 6  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
2019fveq1d 5883 . . . . 5  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
21 ssrab2 3552 . . . . . 6  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  C_  V
22 fvprc 5875 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
231, 22syl5eq 2482 . . . . . 6  |-  ( -.  W  e.  _V  ->  V  =  (/) )
24 sseq0 3800 . . . . . 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 667 . . . . 5  |-  ( -.  W  e.  _V  ->  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  }  =  (/) )
2617, 20, 253eqtr4a 2496 . . . 4  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2726adantr 466 . . 3  |-  ( ( -.  W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2816, 27pm2.61ian 797 . 2  |-  ( S  e.  ~P V  -> 
(  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
294, 28sylbir 216 1  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    C_ wss 3442   (/)c0 3767   ~Pcpw 3985    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   Basecbs 15084  Scalarcsca 15155   .icip 15157   0gc0g 15297   ocvcocv 19154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-ocv 19157
This theorem is referenced by:  elocv  19162  ocv0  19171  csscld  22113  hlhilocv  35237
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