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Theorem ocvi 18205
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 18204 . . 3  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
76simp3bi 1005 . 2  |-  ( A  e.  (  ._|_  `  S
)  ->  A. x  e.  S  ( A  .,  x )  =  .0.  )
8 oveq2 6200 . . . 4  |-  ( x  =  B  ->  ( A  .,  x )  =  ( A  .,  B
) )
98eqeq1d 2453 . . 3  |-  ( x  =  B  ->  (
( A  .,  x
)  =  .0.  <->  ( A  .,  B )  =  .0.  ) )
109rspccva 3170 . 2  |-  ( ( A. x  e.  S  ( A  .,  x )  =  .0.  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )
117, 10sylan 471 1  |-  ( ( A  e.  (  ._|_  `  S )  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3428   ` cfv 5518  (class class class)co 6192   Basecbs 14278  Scalarcsca 14345   .icip 14347   0gc0g 14482   ocvcocv 18196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-ocv 18199
This theorem is referenced by:  ocvocv  18207  ocvlss  18208  ocvin  18210  lsmcss  18228  clsocv  20880
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