Theorem occon 27530

Description: Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
occon	⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Proof of Theorem occon

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

Step	Hyp	Ref	Expression
1		ssralv 3629	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
2	1	ralrimivw 2950	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ ℋ (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
3		ss2rab 3641	. . . . 5 ⊢ ({𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} ↔ ∀𝑥 ∈ ℋ (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
4	2, 3	sylibr 223	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
5	4	adantl 481	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
6		ocval 27523	. . . 4 ⊢ (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
7	6	ad2antlr 759	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
8		ocval 27523	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
9	8	ad2antrr 758	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
10	5, 7, 9	3sstr4d 3611	. 2 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
11	10	ex 449	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∀wral 2896  {crab 2900   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ℋchil 27160   ·_ih csp 27163  ⊥cort 27171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-oc 27493

This theorem is referenced by:  occon2  27531  occon3  27540  ococin  27651  ssjo  27690  chsscon3i  27704  shjshsi  27735