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Theorem ocvfval 16566
Description: The orthocomplement operation. (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
ocvfval  |-  ( W  e.  X  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
Distinct variable groups:    x, s,
y,  .0.    V, s, x, y    W, s, x, y    ., , s, x, y
Allowed substitution hints:    F( x, y, s)    ._|_ ( x, y, s)    X( x, y, s)

Proof of Theorem ocvfval
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 ocvfval.o . 2  |-  ._|_  =  ( ocv `  W )
2 elex 2796 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5525 . . . . . . 7  |-  ( h  =  W  ->  ( Base `  h )  =  ( Base `  W
) )
4 ocvfval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2333 . . . . . 6  |-  ( h  =  W  ->  ( Base `  h )  =  V )
65pweqd 3630 . . . . 5  |-  ( h  =  W  ->  ~P ( Base `  h )  =  ~P V )
7 fveq2 5525 . . . . . . . . . 10  |-  ( h  =  W  ->  ( .i `  h )  =  ( .i `  W
) )
8 ocvfval.i . . . . . . . . . 10  |-  .,  =  ( .i `  W )
97, 8syl6eqr 2333 . . . . . . . . 9  |-  ( h  =  W  ->  ( .i `  h )  = 
.,  )
109oveqd 5875 . . . . . . . 8  |-  ( h  =  W  ->  (
x ( .i `  h ) y )  =  ( x  .,  y ) )
11 fveq2 5525 . . . . . . . . . . 11  |-  ( h  =  W  ->  (Scalar `  h )  =  (Scalar `  W ) )
12 ocvfval.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
1311, 12syl6eqr 2333 . . . . . . . . . 10  |-  ( h  =  W  ->  (Scalar `  h )  =  F )
1413fveq2d 5529 . . . . . . . . 9  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  ( 0g `  F ) )
15 ocvfval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  F )
1614, 15syl6eqr 2333 . . . . . . . 8  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  .0.  )
1710, 16eqeq12d 2297 . . . . . . 7  |-  ( h  =  W  ->  (
( x ( .i
`  h ) y )  =  ( 0g
`  (Scalar `  h )
)  <->  ( x  .,  y )  =  .0.  ) )
1817ralbidv 2563 . . . . . 6  |-  ( h  =  W  ->  ( A. y  e.  s 
( x ( .i
`  h ) y )  =  ( 0g
`  (Scalar `  h )
)  <->  A. y  e.  s  ( x  .,  y
)  =  .0.  )
)
195, 18rabeqbidv 2783 . . . . 5  |-  ( h  =  W  ->  { x  e.  ( Base `  h
)  |  A. y  e.  s  ( x
( .i `  h
) y )  =  ( 0g `  (Scalar `  h ) ) }  =  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} )
206, 19mpteq12dv 4098 . . . 4  |-  ( h  =  W  ->  (
s  e.  ~P ( Base `  h )  |->  { x  e.  ( Base `  h )  |  A. y  e.  s  (
x ( .i `  h ) y )  =  ( 0g `  (Scalar `  h ) ) } )  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y
)  =  .0.  }
) )
21 df-ocv 16563 . . . 4  |-  ocv  =  ( h  e.  _V  |->  ( s  e.  ~P ( Base `  h )  |->  { x  e.  (
Base `  h )  |  A. y  e.  s  ( x ( .i
`  h ) y )  =  ( 0g
`  (Scalar `  h )
) } ) )
22 eqid 2283 . . . . . 6  |-  ( 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.  }
)
23 ssrab2 3258 . . . . . . . 8  |-  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  }  C_  V
24 fvex 5539 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
254, 24eqeltri 2353 . . . . . . . . 9  |-  V  e. 
_V
2625elpw2 4175 . . . . . . . 8  |-  ( { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }  e.  ~P V  <->  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  }  C_  V
)
2723, 26mpbir 200 . . . . . . 7  |-  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  }  e.  ~P V
2827a1i 10 . . . . . 6  |-  ( s  e.  ~P V  ->  { x  e.  V  |  A. y  e.  s  ( x  .,  y
)  =  .0.  }  e.  ~P V )
2922, 28fmpti 5683 . . . . 5  |-  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }
) : ~P V --> ~P V
3025pwex 4193 . . . . 5  |-  ~P V  e.  _V
31 fex2 5401 . . . . 5  |-  ( ( ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) : ~P V
--> ~P V  /\  ~P V  e.  _V  /\  ~P V  e.  _V )  ->  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} )  e.  _V )
3229, 30, 30, 31mp3an 1277 . . . 4  |-  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }
)  e.  _V
3320, 21, 32fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  ( ocv `  W )  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
342, 33syl 15 . 2  |-  ( W  e.  X  ->  ( ocv `  W )  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
351, 34syl5eq 2327 1  |-  ( W  e.  X  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   ocvcocv 16560
This theorem is referenced by:  ocvval  16567  elocv  16568
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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