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Theorem ocvi 21687
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 21686 . . 3 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
76simp3bi 1148 . 2 (𝐴 ∈ ( 𝑆) → ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )
8 oveq2 7439 . . . 4 (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
98eqeq1d 2739 . . 3 (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
109rspccva 3621 . 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 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300  ·𝑖cip 17302  0gc0g 17484  ocvcocv 21678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-ocv 21681
This theorem is referenced by:  ocvocv  21689  ocvlss  21690  ocvin  21692  lsmcss  21710  clsocv  25284
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