Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocv2ss Structured version   Visualization version   GIF version

Theorem ocv2ss 19836
 Description: Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypothesis
Ref Expression
ocv2ss.o = (ocv‘𝑊)
Assertion
Ref Expression
ocv2ss (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))

Proof of Theorem ocv2ss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr2 3575 . . . 4 (𝑇𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2 idd 24 . . . 4 (𝑇𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3 ssralv 3629 . . . 4 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)) → ∀𝑦𝑇 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
41, 2, 33anim123d 1398 . . 3 (𝑇𝑆 → ((𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑆 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) → (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑇 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
5 eqid 2610 . . . 4 (Base‘𝑊) = (Base‘𝑊)
6 eqid 2610 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
7 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2610 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9 ocv2ss.o . . . 4 = (ocv‘𝑊)
105, 6, 7, 8, 9elocv 19831 . . 3 (𝑥 ∈ ( 𝑆) ↔ (𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑆 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
115, 6, 7, 8, 9elocv 19831 . . 3 (𝑥 ∈ ( 𝑇) ↔ (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑇 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
124, 10, 113imtr4g 284 . 2 (𝑇𝑆 → (𝑥 ∈ ( 𝑆) → 𝑥 ∈ ( 𝑇)))
1312ssrdv 3574 1 (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771  ·𝑖cip 15773  0gc0g 15923  ocvcocv 19823 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-ocv 19826 This theorem is referenced by:  ocvsscon  19838  ocvlsp  19839  ocvcss  19850  cssmre  19856  mrccss  19857  clsocv  22857
 Copyright terms: Public domain W3C validator