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Theorem ocvfval 18823
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 3118 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5872 . . . . . . 7  |-  ( h  =  W  ->  ( Base `  h )  =  ( Base `  W
) )
4 ocvfval.v . . . . . . 7  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2516 . . . . . 6  |-  ( h  =  W  ->  ( Base `  h )  =  V )
65pweqd 4020 . . . . 5  |-  ( h  =  W  ->  ~P ( Base `  h )  =  ~P V )
7 fveq2 5872 . . . . . . . . . 10  |-  ( h  =  W  ->  ( .i `  h )  =  ( .i `  W
) )
8 ocvfval.i . . . . . . . . . 10  |-  .,  =  ( .i `  W )
97, 8syl6eqr 2516 . . . . . . . . 9  |-  ( h  =  W  ->  ( .i `  h )  = 
.,  )
109oveqd 6313 . . . . . . . 8  |-  ( h  =  W  ->  (
x ( .i `  h ) y )  =  ( x  .,  y ) )
11 fveq2 5872 . . . . . . . . . . 11  |-  ( h  =  W  ->  (Scalar `  h )  =  (Scalar `  W ) )
12 ocvfval.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
1311, 12syl6eqr 2516 . . . . . . . . . 10  |-  ( h  =  W  ->  (Scalar `  h )  =  F )
1413fveq2d 5876 . . . . . . . . 9  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  ( 0g `  F ) )
15 ocvfval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  F )
1614, 15syl6eqr 2516 . . . . . . . 8  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  .0.  )
1710, 16eqeq12d 2479 . . . . . . 7  |-  ( h  =  W  ->  (
( x ( .i
`  h ) y )  =  ( 0g
`  (Scalar `  h )
)  <->  ( x  .,  y )  =  .0.  ) )
1817ralbidv 2896 . . . . . 6  |-  ( h  =  W  ->  ( A. y  e.  s 
( x ( .i
`  h ) y )  =  ( 0g
`  (Scalar `  h )
)  <->  A. y  e.  s  ( x  .,  y
)  =  .0.  )
)
195, 18rabeqbidv 3104 . . . . 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 4535 . . . 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 18820 . . . 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 2457 . . . . . 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 3581 . . . . . . . 8  |-  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  }  C_  V
24 fvex 5882 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
254, 24eqeltri 2541 . . . . . . . . 9  |-  V  e. 
_V
2625elpw2 4620 . . . . . . . 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 209 . . . . . . 7  |-  { x  e.  V  |  A. y  e.  s  (
x  .,  y )  =  .0.  }  e.  ~P V
2827a1i 11 . . . . . 6  |-  ( s  e.  ~P V  ->  { x  e.  V  |  A. y  e.  s  ( x  .,  y
)  =  .0.  }  e.  ~P V )
2922, 28fmpti 6055 . . . . 5  |-  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }
) : ~P V --> ~P V
3025pwex 4639 . . . . 5  |-  ~P V  e.  _V
31 fex2 6754 . . . . 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 1324 . . . 4  |-  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s 
( x  .,  y
)  =  .0.  }
)  e.  _V
3320, 21, 32fvmpt 5956 . . 3  |-  ( W  e.  _V  ->  ( ocv `  W )  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
342, 33syl 16 . 2  |-  ( W  e.  X  ->  ( ocv `  W )  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
351, 34syl5eq 2510 1  |-  ( W  e.  X  ->  ._|_  =  ( s  e.  ~P V  |->  { x  e.  V  |  A. y  e.  s  ( x  .,  y )  =  .0. 
} ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14643  Scalarcsca 14714   .icip 14716   0gc0g 14856   ocvcocv 18817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-ocv 18820
This theorem is referenced by:  ocvval  18824  elocv  18825
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