Theorem ocv2ss 21598

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 3944	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2		idd 24	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3		ssralv 4006	. . . 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 1445	. . 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 2729	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
6		eqid 2729	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
7		eqid 2729	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2729	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9		ocv2ss.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
10	5, 6, 7, 8, 9	elocv 21593	. . 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 21593	. . 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 3943	1 ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3905  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Scalarcsca 17182  ·_𝑖cip 17184  0_gc0g 17361  ocvcocv 21585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-ocv 21588

This theorem is referenced by:  ocvsscon  21600  ocvlsp  21601  ocvcss  21612  cssmre  21618  mrccss  21619  clsocv  25166