MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocvi Structured version   Visualization version   GIF version

Theorem ocvi 21705
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 21704 . . 3 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
76simp3bi 1146 . 2 (𝐴 ∈ ( 𝑆) → ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )
8 oveq2 7439 . . . 4 (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
98eqeq1d 2737 . . 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 1537  wcel 2106  wral 3059  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  ocvcocv 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-ocv 21699
This theorem is referenced by:  ocvocv  21707  ocvlss  21708  ocvin  21710  lsmcss  21728  clsocv  25298
  Copyright terms: Public domain W3C validator