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	⊢ (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 3985	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ (Base‘𝑊) → 𝑇 ⊆ (Base‘𝑊)))
2		idd 24	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → (𝑥 ∈ (Base‘𝑊) → 𝑥 ∈ (Base‘𝑊)))
3		ssralv 4046	. . . 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 1440	. . 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 2727	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
6		eqid 2727	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
7		eqid 2727	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2727	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
9		ocv2ss.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
10	5, 6, 7, 8, 9	elocv 21580	. . 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 21580	. . 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 3984	1 ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇))

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  0_gc0g 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