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Theorem elocv 16568
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 5554 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  dom  ._|_  )
2 n0i 3460 . . . . . . . . 9  |-  ( A  e.  (  ._|_  `  S
)  ->  -.  (  ._|_  `  S )  =  (/) )
3 ocvfval.o . . . . . . . . . . . 12  |-  ._|_  =  ( ocv `  W )
4 fvprc 5519 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
53, 4syl5eq 2327 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
65fveq1d 5527 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
7 fv01 5559 . . . . . . . . . 10  |-  ( (/) `  S )  =  (/)
86, 7syl6eq 2331 . . . . . . . . 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 16566 . . . . . . . 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 4881 . . . . . 6  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  dom  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
17 fvex 5539 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1810, 17eqeltri 2353 . . . . . . . 8  |-  V  e. 
_V
1918rabex 4165 . . . . . . 7  |-  { y  e.  V  |  A. x  e.  s  (
y  .,  x )  =  .0.  }  e.  _V
20 eqid 2283 . . . . . . 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 5373 . . . . . 6  |-  dom  (
s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
)  =  ~P V
2216, 21syl6eq 2331 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  ~P V )
231, 22eleqtrd 2359 . . . 4  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  ~P V )
24 elpwi 3633 . . . 4  |-  ( S  e.  ~P V  ->  S  C_  V )
2523, 24syl 15 . . 3  |-  ( A  e.  (  ._|_  `  S
)  ->  S  C_  V
)
2610, 11, 12, 13, 3ocvval 16567 . . . . 5  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
)
2726eleq2d 2350 . . . 4  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
) )
28 oveq1 5865 . . . . . . 7  |-  ( y  =  A  ->  (
y  .,  x )  =  ( A  .,  x ) )
2928eqeq1d 2291 . . . . . 6  |-  ( y  =  A  ->  (
( y  .,  x
)  =  .0.  <->  ( A  .,  x )  =  .0.  ) )
3029ralbidv 2563 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  S  ( y  .,  x
)  =  .0.  <->  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3130elrab 2923 . . . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   ocvcocv 16560
This theorem is referenced by:  ocvi  16569  ocvss  16570  ocvocv  16571  ocvlss  16572  ocv2ss  16573  unocv  16580  iunocv  16581  obselocv  16628  clsocv  18677  pjthlem2  18802
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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