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Theorem elocv 19168
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 5898 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  dom  ._|_  )
2 n0i 3763 . . . . . . . . 9  |-  ( A  e.  (  ._|_  `  S
)  ->  -.  (  ._|_  `  S )  =  (/) )
3 ocvfval.o . . . . . . . . . . . 12  |-  ._|_  =  ( ocv `  W )
4 fvprc 5866 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
53, 4syl5eq 2473 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
65fveq1d 5874 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
7 0fv 5905 . . . . . . . . . 10  |-  ( (/) `  S )  =  (/)
86, 7syl6eq 2477 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  (/) )
92, 8nsyl2 130 . . . . . . . 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 19166 . . . . . . . 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 5048 . . . . . 6  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  dom  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
17 fvex 5882 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1810, 17eqeltri 2504 . . . . . . . 8  |-  V  e. 
_V
1918rabex 4567 . . . . . . 7  |-  { y  e.  V  |  A. x  e.  s  (
y  .,  x )  =  .0.  }  e.  _V
20 eqid 2420 . . . . . . 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 5716 . . . . . 6  |-  dom  (
s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
)  =  ~P V
2216, 21syl6eq 2477 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  ~P V )
231, 22eleqtrd 2510 . . . 4  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  ~P V )
2423elpwid 3986 . . 3  |-  ( A  e.  (  ._|_  `  S
)  ->  S  C_  V
)
2510, 11, 12, 13, 3ocvval 19167 . . . . 5  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
)
2625eleq2d 2490 . . . 4  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
) )
27 oveq1 6303 . . . . . . 7  |-  ( y  =  A  ->  (
y  .,  x )  =  ( A  .,  x ) )
2827eqeq1d 2422 . . . . . 6  |-  ( y  =  A  ->  (
( y  .,  x
)  =  .0.  <->  ( A  .,  x )  =  .0.  ) )
2928ralbidv 2862 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  S  ( y  .,  x
)  =  .0.  <->  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3029elrab 3226 . . . 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 264 . . 3  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
3224, 31biadan2 646 . 2  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
33 3anass 986 . 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 255 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   {crab 2777   _Vcvv 3078    C_ wss 3433   (/)c0 3758   ~Pcpw 3976    |-> cmpt 4475   dom cdm 4845   ` cfv 5592  (class class class)co 6296   Basecbs 15081  Scalarcsca 15153   .icip 15155   0gc0g 15298   ocvcocv 19160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-ocv 19163
This theorem is referenced by:  ocvi  19169  ocvss  19170  ocvocv  19171  ocvlss  19172  ocv2ss  19173  unocv  19180  iunocv  19181  obselocv  19228  clsocv  22140  pjthlem2  22299
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