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Theorem elocv 16812
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 5691 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  dom  ._|_  )
2 n0i 3570 . . . . . . . . 9  |-  ( A  e.  (  ._|_  `  S
)  ->  -.  (  ._|_  `  S )  =  (/) )
3 ocvfval.o . . . . . . . . . . . 12  |-  ._|_  =  ( ocv `  W )
4 fvprc 5656 . . . . . . . . . . . 12  |-  ( -.  W  e.  _V  ->  ( ocv `  W )  =  (/) )
53, 4syl5eq 2425 . . . . . . . . . . 11  |-  ( -.  W  e.  _V  ->  ._|_ 
=  (/) )
65fveq1d 5664 . . . . . . . . . 10  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  ( (/) `  S ) )
7 fv01 5696 . . . . . . . . . 10  |-  ( (/) `  S )  =  (/)
86, 7syl6eq 2429 . . . . . . . . 9  |-  ( -.  W  e.  _V  ->  ( 
._|_  `  S )  =  (/) )
92, 8nsyl2 121 . . . . . . . 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 16810 . . . . . . . 8  |-  ( W  e.  _V  ->  ._|_  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
159, 14syl 16 . . . . . . 7  |-  ( A  e.  (  ._|_  `  S
)  ->  ._|_  =  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
) )
1615dmeqd 5006 . . . . . 6  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  dom  ( s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x )  =  .0. 
} ) )
17 fvex 5676 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
1810, 17eqeltri 2451 . . . . . . . 8  |-  V  e. 
_V
1918rabex 4289 . . . . . . 7  |-  { y  e.  V  |  A. x  e.  s  (
y  .,  x )  =  .0.  }  e.  _V
20 eqid 2381 . . . . . . 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 5508 . . . . . 6  |-  dom  (
s  e.  ~P V  |->  { y  e.  V  |  A. x  e.  s  ( y  .,  x
)  =  .0.  }
)  =  ~P V
2216, 21syl6eq 2429 . . . . 5  |-  ( A  e.  (  ._|_  `  S
)  ->  dom  ._|_  =  ~P V )
231, 22eleqtrd 2457 . . . 4  |-  ( A  e.  (  ._|_  `  S
)  ->  S  e.  ~P V )
2423elpwid 3745 . . 3  |-  ( A  e.  (  ._|_  `  S
)  ->  S  C_  V
)
2510, 11, 12, 13, 3ocvval 16811 . . . . 5  |-  ( S 
C_  V  ->  (  ._|_  `  S )  =  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
)
2625eleq2d 2448 . . . 4  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  A  e.  { y  e.  V  |  A. x  e.  S  ( y  .,  x
)  =  .0.  }
) )
27 oveq1 6021 . . . . . . 7  |-  ( y  =  A  ->  (
y  .,  x )  =  ( A  .,  x ) )
2827eqeq1d 2389 . . . . . 6  |-  ( y  =  A  ->  (
( y  .,  x
)  =  .0.  <->  ( A  .,  x )  =  .0.  ) )
2928ralbidv 2663 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  S  ( y  .,  x
)  =  .0.  <->  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
3029elrab 3029 . . . 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 253 . . 3  |-  ( S 
C_  V  ->  ( A  e.  (  ._|_  `  S )  <->  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
3224, 31biadan2 624 . 2  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  ( A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) ) )
33 3anass 940 . 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 244 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2643   {crab 2647   _Vcvv 2893    C_ wss 3257   (/)c0 3565   ~Pcpw 3736    e. cmpt 4201   dom cdm 4812   ` cfv 5388  (class class class)co 6014   Basecbs 13390  Scalarcsca 13453   .icip 13455   0gc0g 13644   ocvcocv 16804
This theorem is referenced by:  ocvi  16813  ocvss  16814  ocvocv  16815  ocvlss  16816  ocv2ss  16817  unocv  16824  iunocv  16825  obselocv  16872  clsocv  19069  pjthlem2  19200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-sbc 3099  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-fv 5396  df-ov 6017  df-ocv 16807
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