Theorem ocv2ss 20859

Description: Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypothesis

Ref	Expression
ocv2ss.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
ocv2ss	⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))

Proof of Theorem ocv2ss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3932	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2		idd 24	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3		ssralv 3991	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) → ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
4	1, 2, 3	3anim123d 1441	. . 3 ⊢ (𝑇 ⊆ 𝑆 → ((𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) → (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
5		eqid 2739	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
6		eqid 2739	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
7		eqid 2739	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2739	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9		ocv2ss.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
10	5, 6, 7, 8, 9	elocv 20854	. . 3 ⊢ (𝑥 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑆 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
11	5, 6, 7, 8, 9	elocv 20854	. . 3 ⊢ (𝑥 ∈ ( ⊥ ‘𝑇) ↔ (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
12	4, 10, 11	3imtr4g 295	. 2 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ ( ⊥ ‘𝑆) → 𝑥 ∈ ( ⊥ ‘𝑇)))
13	12	ssrdv 3931	1 ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065   ⊆ wss 3891  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  Scalarcsca 16946  ·_𝑖cip 16948  0_gc0g 17131  ocvcocv 20846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-ov 7271  df-ocv 20849

This theorem is referenced by:  ocvsscon  20861  ocvlsp  20862  ocvcss  20873  cssmre  20879  mrccss  20880  clsocv  24395