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Theorem ocvval 16894
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 5742 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2506 . . 3  |-  V  e. 
_V
43elpw2 4364 . 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 16893 . . . . 5  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
109fveq1d 5730 . . . 4  |-  ( W  e.  _V  ->  (  ._|_  `  S )  =  ( ( s  e. 
~P V  |->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  } ) `  S ) )
11 raleq 2904 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .,  y
)  =  .0.  <->  A. y  e.  S  ( x  .,  y )  =  .0.  ) )
1211rabbidv 2948 . . . . 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 2436 . . . . 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 4354 . . . . 5  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  e.  _V
1512, 13, 14fvmpt 5806 . . . 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 2488 . . 3  |-  ( ( W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
17 fv01 5763 . . . . 5  |-  ( (/) `  S )  =  (/)
18 fvprc 5722 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
198, 18syl5eq 2480 . . . . . 6  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
2019fveq1d 5730 . . . . 5  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
21 ssrab2 3428 . . . . . 6  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  C_  V
22 fvprc 5722 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
231, 22syl5eq 2480 . . . . . 6  |-  ( -.  W  e.  _V  ->  V  =  (/) )
24 sseq0 3659 . . . . . 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 645 . . . . 5  |-  ( -.  W  e.  _V  ->  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  }  =  (/) )
2617, 20, 253eqtr4a 2494 . . . 4  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2726adantr 452 . . 3  |-  ( ( -.  W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2816, 27pm2.61ian 766 . 2  |-  ( S  e.  ~P V  -> 
(  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
294, 28sylbir 205 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 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .icip 13534   0gc0g 13723   ocvcocv 16887
This theorem is referenced by:  elocv  16895  ocv0  16904  csscld  19203  hlhilocv  32758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-ocv 16890
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