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Theorem ocvval 18493
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 5876 . . . 4  |-  ( Base `  W )  e.  _V
31, 2eqeltri 2551 . . 3  |-  V  e. 
_V
43elpw2 4611 . 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 18492 . . . . 5  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
109fveq1d 5868 . . . 4  |-  ( W  e.  _V  ->  (  ._|_  `  S )  =  ( ( s  e. 
~P V  |->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  } ) `  S ) )
11 raleq 3058 . . . . . 6  |-  ( s  =  S  ->  ( A. y  e.  s 
( x  .,  y
)  =  .0.  <->  A. y  e.  S  ( x  .,  y )  =  .0.  ) )
1211rabbidv 3105 . . . . 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 2467 . . . . 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 4598 . . . . 5  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  e.  _V
1512, 13, 14fvmpt 5950 . . . 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 2528 . . 3  |-  ( ( W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
17 0fv 5899 . . . . 5  |-  ( (/) `  S )  =  (/)
18 fvprc 5860 . . . . . . 7  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
198, 18syl5eq 2520 . . . . . 6  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
2019fveq1d 5868 . . . . 5  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
21 ssrab2 3585 . . . . . 6  |-  { x  e.  V  |  A. y  e.  S  (
x  .,  y )  =  .0.  }  C_  V
22 fvprc 5860 . . . . . . 7  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
231, 22syl5eq 2520 . . . . . 6  |-  ( -.  W  e.  _V  ->  V  =  (/) )
24 sseq0 3817 . . . . . 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 663 . . . . 5  |-  ( -.  W  e.  _V  ->  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  }  =  (/) )
2617, 20, 253eqtr4a 2534 . . . 4  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2726adantr 465 . . 3  |-  ( ( -.  W  e.  _V  /\  S  e.  ~P V
)  ->  (  ._|_  `  S )  =  {
x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
2816, 27pm2.61ian 788 . 2  |-  ( S  e.  ~P V  -> 
(  ._|_  `  S )  =  { x  e.  V  |  A. y  e.  S  ( x  .,  y )  =  .0.  } )
294, 28sylbir 213 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 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   Basecbs 14490  Scalarcsca 14558   .icip 14560   0gc0g 14695   ocvcocv 18486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-ocv 18489
This theorem is referenced by:  elocv  18494  ocv0  18503  csscld  21452  hlhilocv  36775
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