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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) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Proof of Theorem occon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3629 . . . . . 6 (𝐴𝐵 → (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
21ralrimivw 2950 . . . . 5 (𝐴𝐵 → ∀𝑥 ∈ ℋ (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
3 ss2rab 3641 . . . . 5 ({𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0} ↔ ∀𝑥 ∈ ℋ (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
42, 3sylibr 223 . . . 4 (𝐴𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
54adantl 481 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
6 ocval 27523 . . . 4 (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
76ad2antlr 759 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
8 ocval 27523 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
98ad2antrr 758 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
105, 7, 93sstr4d 3611 . 2 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
1110ex 449 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))
 Colors of variables: wff setvar class 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
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