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Theorem elocv 19224
Description: Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed 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
elocv  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
Distinct variable groups:    x,  .0.    x, A    x, V    x, W    x,  .,    x, S
Allowed substitution hints:    F( x)    ._|_ ( x)

Proof of Theorem elocv
Dummy variables  s 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5889 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  dom  ._|_  )
2 n0i 3735 . . . . . . . . 9  |-  ( A  e.  (  ._|_  `  S
)  ->  -.  (  ._|_  `  S )  =  (/) )
3 ocvfval.o . . . . . . . . . . . 12  |-  ._|_  =  ( ocv `  W )
4 fvprc 5857 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
53, 4syl5eq 2496 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
65fveq1d 5865 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
7 0fv 5896 . . . . . . . . . 10  |-  ( (/) `  S )  =  (/)
86, 7syl6eq 2500 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  (/) )
92, 8nsyl2 131 . . . . . . . 8  |-  ( A  e.  (  ._|_  `  S
)  ->  W  e.  _V )
10 ocvfval.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
11 ocvfval.i . . . . . . . . 9  |-  .,  =  ( .i `  W )
12 ocvfval.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
13 ocvfval.z . . . . . . . . 9  |-  .0.  =  ( 0g `  F )
1410, 11, 12, 13, 3ocvfval 19222 . . . . . . . 8  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
159, 14syl 17 . . . . . . 7  |-  ( A  e.  (  ._|_  `  S
)  ->  ._|_  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
) )
1615dmeqd 5036 . . . . . 6  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  dom  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
17 fvex 5873 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1810, 17eqeltri 2524 . . . . . . . 8  |-  V  e. 
_V
1918rabex 4553 . . . . . . 7  |-  { y  e.  V  |  A. x  e.  s  (
y  .,  x )  =  .0.  }  e.  _V
20 eqid 2450 . . . . . . 7  |-  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s 
( y  .,  x
)  =  .0.  }
)  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s 
( y  .,  x
)  =  .0.  }
)
2119, 20dmmpti 5705 . . . . . 6  |-  dom  (
s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
)  =  ~P V
2216, 21syl6eq 2500 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  ~P V )
231, 22eleqtrd 2530 . . . 4  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  ~P V )
2423elpwid 3960 . . 3  |-  ( A  e.  (  ._|_  `  S
)  ->  S  C_  V
)
2510, 11, 12, 13, 3ocvval 19223 . . . . 5  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
)
2625eleq2d 2513 . . . 4  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
) )
27 oveq1 6295 . . . . . . 7  |-  ( y  =  A  ->  (
y  .,  x )  =  ( A  .,  x ) )
2827eqeq1d 2452 . . . . . 6  |-  ( y  =  A  ->  (
( y  .,  x
)  =  .0.  <->  ( A  .,  x )  =  .0.  ) )
2928ralbidv 2826 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  S  ( y  .,  x
)  =  .0.  <->  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3029elrab 3195 . . . 4  |-  ( A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x )  =  .0. 
}  <->  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3126, 30syl6bb 265 . . 3  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
3224, 31biadan2 647 . 2  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
33 3anass 988 . 2  |-  ( ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  )  <->  ( S  C_  V  /\  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
3432, 33bitr4i 256 1  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   {crab 2740   _Vcvv 3044    C_ wss 3403   (/)c0 3730   ~Pcpw 3950    |-> cmpt 4460   dom cdm 4833   ` cfv 5581  (class class class)co 6288   Basecbs 15114  Scalarcsca 15186   .icip 15188   0gc0g 15331   ocvcocv 19216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6291  df-ocv 19219
This theorem is referenced by:  ocvi  19225  ocvss  19226  ocvocv  19227  ocvlss  19228  ocv2ss  19229  unocv  19236  iunocv  19237  obselocv  19284  clsocv  22214  pjthlem2  22385
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