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Theorem ocvi 16898
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 16897 . . 3  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
76simp3bi 975 . 2  |-  ( A  e.  (  ._|_  `  S
)  ->  A. x  e.  S  ( A  .,  x )  =  .0.  )
8 oveq2 6091 . . . 4  |-  ( x  =  B  ->  ( A  .,  x )  =  ( A  .,  B
) )
98eqeq1d 2446 . . 3  |-  ( x  =  B  ->  (
( A  .,  x
)  =  .0.  <->  ( A  .,  B )  =  .0.  ) )
109rspccva 3053 . 2  |-  ( ( A. x  e.  S  ( A  .,  x )  =  .0.  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )
117, 10sylan 459 1  |-  ( ( A  e.  (  ._|_  `  S )  /\  B  e.  S )  ->  ( A  .,  B )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .icip 13536   0gc0g 13725   ocvcocv 16889
This theorem is referenced by:  ocvocv  16900  ocvlss  16901  ocvin  16903  lsmcss  16921  clsocv  19206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-ocv 16892
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