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Theorem elocv 16584
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 5570 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  dom  ._|_  )
2 n0i 3473 . . . . . . . . 9  |-  ( A  e.  (  ._|_  `  S
)  ->  -.  (  ._|_  `  S )  =  (/) )
3 ocvfval.o . . . . . . . . . . . 12  |-  ._|_  =  ( ocv `  W )
4 fvprc 5535 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
53, 4syl5eq 2340 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
65fveq1d 5543 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
7 fv01 5575 . . . . . . . . . 10  |-  ( (/) `  S )  =  (/)
86, 7syl6eq 2344 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  (/) )
92, 8nsyl2 119 . . . . . . . 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 16582 . . . . . . . 8  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
159, 14syl 15 . . . . . . 7  |-  ( A  e.  (  ._|_  `  S
)  ->  ._|_  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
) )
1615dmeqd 4897 . . . . . 6  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  dom  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
17 fvex 5555 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1810, 17eqeltri 2366 . . . . . . . 8  |-  V  e. 
_V
1918rabex 4181 . . . . . . 7  |-  { y  e.  V  |  A. x  e.  s  (
y  .,  x )  =  .0.  }  e.  _V
20 eqid 2296 . . . . . . 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 5389 . . . . . 6  |-  dom  (
s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
)  =  ~P V
2216, 21syl6eq 2344 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  ~P V )
231, 22eleqtrd 2372 . . . 4  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  ~P V )
24 elpwi 3646 . . . 4  |-  ( S  e.  ~P V  ->  S  C_  V )
2523, 24syl 15 . . 3  |-  ( A  e.  (  ._|_  `  S
)  ->  S  C_  V
)
2610, 11, 12, 13, 3ocvval 16583 . . . . 5  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
)
2726eleq2d 2363 . . . 4  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
) )
28 oveq1 5881 . . . . . . 7  |-  ( y  =  A  ->  (
y  .,  x )  =  ( A  .,  x ) )
2928eqeq1d 2304 . . . . . 6  |-  ( y  =  A  ->  (
( y  .,  x
)  =  .0.  <->  ( A  .,  x )  =  .0.  ) )
3029ralbidv 2576 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  S  ( y  .,  x
)  =  .0.  <->  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3130elrab 2936 . . . 4  |-  ( A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x )  =  .0. 
}  <->  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3227, 31syl6bb 252 . . 3  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
3325, 32biadan2 623 . 2  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
34 3anass 938 . 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.  ) ) )
3533, 34bitr4i 243 1  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .icip 13229   0gc0g 13416   ocvcocv 16576
This theorem is referenced by:  ocvi  16585  ocvss  16586  ocvocv  16587  ocvlss  16588  ocv2ss  16589  unocv  16596  iunocv  16597  obselocv  16644  clsocv  18693  pjthlem2  18818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-ocv 16579
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