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Theorem ocv2ss 21585
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 3985 . . . 4 (𝑇𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2 idd 24 . . . 4 (𝑇𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3 ssralv 4046 . . . 4 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)) → ∀𝑦𝑇 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
41, 2, 33anim123d 1440 . . 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 2727 . . . 4 (Base‘𝑊) = (Base‘𝑊)
6 eqid 2727 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
7 eqid 2727 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2727 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9 ocv2ss.o . . . 4 = (ocv‘𝑊)
105, 6, 7, 8, 9elocv 21580 . . 3 (𝑥 ∈ ( 𝑆) ↔ (𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑆 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
115, 6, 7, 8, 9elocv 21580 . . 3 (𝑥 ∈ ( 𝑇) ↔ (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑇 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
124, 10, 113imtr4g 296 . 2 (𝑇𝑆 → (𝑥 ∈ ( 𝑆) → 𝑥 ∈ ( 𝑇)))
1312ssrdv 3984 1 (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  wral 3056  wss 3944  cfv 6542  (class class class)co 7414  Basecbs 17165  Scalarcsca 17221  ·𝑖cip 17223  0gc0g 17406  ocvcocv 21572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-ocv 21575
This theorem is referenced by:  ocvsscon  21587  ocvlsp  21588  ocvcss  21599  cssmre  21605  mrccss  21606  clsocv  25152
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