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Theorem occon 30505
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 4048 . . . . . 6 (𝐴𝐵 → (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
21adantr 482 . . . . 5 ((𝐴𝐵𝑥 ∈ ℋ) → (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
32ss2rabdv 4071 . . . 4 (𝐴𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
43adantl 483 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
5 ocval 30498 . . . 4 (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
65ad2antlr 726 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
7 ocval 30498 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
87ad2antrr 725 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
94, 6, 83sstr4d 4027 . 2 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
109ex 414 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wss 3946  cfv 6535  (class class class)co 7396  0cc0 11097  chba 30137   ·ih csp 30140  cort 30148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-hilex 30217
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-oc 30470
This theorem is referenced by:  occon2  30506  occon3  30515  ococin  30626  ssjo  30665  chsscon3i  30679  shjshsi  30710
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