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Theorem ocv2ss 20793
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 3950 . . . 4 (𝑇𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2 idd 24 . . . 4 (𝑇𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3 ssralv 4009 . . . 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 2821 . . . 4 (Base‘𝑊) = (Base‘𝑊)
6 eqid 2821 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
7 eqid 2821 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
8 eqid 2821 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9 ocv2ss.o . . . 4 = (ocv‘𝑊)
105, 6, 7, 8, 9elocv 20788 . . 3 (𝑥 ∈ ( 𝑆) ↔ (𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑆 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
115, 6, 7, 8, 9elocv 20788 . . 3 (𝑥 ∈ ( 𝑇) ↔ (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦𝑇 (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
124, 10, 113imtr4g 299 . 2 (𝑇𝑆 → (𝑥 ∈ ( 𝑆) → 𝑥 ∈ ( 𝑇)))
1312ssrdv 3949 1 (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wral 3126  wss 3910  cfv 6328  (class class class)co 7130  Basecbs 16462  Scalarcsca 16547  ·𝑖cip 16549  0gc0g 16692  ocvcocv 20780
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-ocv 20783
This theorem is referenced by:  ocvsscon  20795  ocvlsp  20796  ocvcss  20807  cssmre  20813  mrccss  20814  clsocv  23833
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