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Theorem ocvi 21213
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 , = (·𝑖𝑊)
ocvfval.f 𝐹 = (Scalar‘𝑊)
ocvfval.z 0 = (0g𝐹)
ocvfval.o = (ocv‘𝑊)
Assertion
Ref Expression
ocvi ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (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 , = (·𝑖𝑊)
3 ocvfval.f . . . 4 𝐹 = (Scalar‘𝑊)
4 ocvfval.z . . . 4 0 = (0g𝐹)
5 ocvfval.o . . . 4 = (ocv‘𝑊)
61, 2, 3, 4, 5elocv 21212 . . 3 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
76simp3bi 1147 . 2 (𝐴 ∈ ( 𝑆) → ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )
8 oveq2 7413 . . . 4 (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
98eqeq1d 2734 . . 3 (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
109rspccva 3611 . 2 ((∀𝑥𝑆 (𝐴 , 𝑥) = 0𝐵𝑆) → (𝐴 , 𝐵) = 0 )
117, 10sylan 580 1 ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (𝐴 , 𝐵) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  wss 3947  cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196  ·𝑖cip 17198  0gc0g 17381  ocvcocv 21204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-ocv 21207
This theorem is referenced by:  ocvocv  21215  ocvlss  21216  ocvin  21218  lsmcss  21236  clsocv  24758
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