Theorem ocv2ss 21644

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 3970	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2		idd 24	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3		ssralv 4032	. . . 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 1444	. . 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 2734	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
6		eqid 2734	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
7		eqid 2734	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2734	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9		ocv2ss.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
10	5, 6, 7, 8, 9	elocv 21639	. . 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 21639	. . 3 ⊢ (𝑥 ∈ ( ⊥ ‘𝑇) ↔ (𝑇 ⊆ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝑇 (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
12	4, 10, 11	3imtr4g 296	. 2 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ ( ⊥ ‘𝑆) → 𝑥 ∈ ( ⊥ ‘𝑇)))
13	12	ssrdv 3969	1 ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050   ⊆ wss 3931  ‘cfv 6540  (class class class)co 7412  Basecbs 17228  Scalarcsca 17275  ·_𝑖cip 17277  0_gc0g 17454  ocvcocv 21631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7415  df-ocv 21634

This theorem is referenced by:  ocvsscon  21646  ocvlsp  21647  ocvcss  21658  cssmre  21664  mrccss  21665  clsocv  25219