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Theorem ocvi 16569
Description: Property of a member of the orthocomplement of a subset. (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
ocvi  |-  ( ( A  e.  (  ._|_  `  S )  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )

Proof of Theorem ocvi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ocvfval.v . . . 4  |-  V  =  ( Base `  W
)
2 ocvfval.i . . . 4  |-  .,  =  ( .i `  W )
3 ocvfval.f . . . 4  |-  F  =  (Scalar `  W )
4 ocvfval.z . . . 4  |-  .0.  =  ( 0g `  F )
5 ocvfval.o . . . 4  |-  ._|_  =  ( ocv `  W )
61, 2, 3, 4, 5elocv 16568 . . 3  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
76simp3bi 972 . 2  |-  ( A  e.  (  ._|_  `  S
)  ->  A. x  e.  S  ( A  .,  x )  =  .0.  )
8 oveq2 5866 . . . 4  |-  ( x  =  B  ->  ( A  .,  x )  =  ( A  .,  B
) )
98eqeq1d 2291 . . 3  |-  ( x  =  B  ->  (
( A  .,  x
)  =  .0.  <->  ( A  .,  B )  =  .0.  ) )
109rspccva 2883 . 2  |-  ( ( A. x  e.  S  ( A  .,  x )  =  .0.  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )
117, 10sylan 457 1  |-  ( ( A  e.  (  ._|_  `  S )  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   ocvcocv 16560
This theorem is referenced by:  ocvocv  16571  ocvlss  16572  ocvin  16574  lsmcss  16592  clsocv  18677
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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