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Theorem occon 28034
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 3651 . . . . . 6 (𝐴𝐵 → (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
21ralrimivw 2963 . . . . 5 (𝐴𝐵 → ∀𝑥 ∈ ℋ (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
3 ss2rab 3663 . . . . 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 224 . . . 4 (𝐴𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
54adantl 482 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
6 ocval 28027 . . . 4 (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
76ad2antlr 762 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
8 ocval 28027 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
98ad2antrr 761 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
105, 7, 93sstr4d 3633 . 2 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
1110ex 450 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wral 2908  {crab 2912  wss 3560  cfv 5857  (class class class)co 6615  0cc0 9896  chil 27664   ·ih csp 27667  cort 27675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-hilex 27744
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-oc 27997
This theorem is referenced by:  occon2  28035  occon3  28044  ococin  28155  ssjo  28194  chsscon3i  28208  shjshsi  28239
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