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Theorem ocvi 18467
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 18466 . . 3  |-  ( A  e.  (  ._|_  `  S
)  <->  ( S  C_  V  /\  A  e.  V  /\  A. x  e.  S  ( A  .,  x )  =  .0.  ) )
76simp3bi 1013 . 2  |-  ( A  e.  (  ._|_  `  S
)  ->  A. x  e.  S  ( A  .,  x )  =  .0.  )
8 oveq2 6290 . . . 4  |-  ( x  =  B  ->  ( A  .,  x )  =  ( A  .,  B
) )
98eqeq1d 2469 . . 3  |-  ( x  =  B  ->  (
( A  .,  x
)  =  .0.  <->  ( A  .,  B )  =  .0.  ) )
109rspccva 3213 . 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 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   ` cfv 5586  (class class class)co 6282   Basecbs 14486  Scalarcsca 14554   .icip 14556   0gc0g 14691   ocvcocv 18458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-ocv 18461
This theorem is referenced by:  ocvocv  18469  ocvlss  18470  ocvin  18472  lsmcss  18490  clsocv  21425
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