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Theorem occon 29049
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 4012 . . . . . 6 (𝐴𝐵 → (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
21ralrimivw 3170 . . . . 5 (𝐴𝐵 → ∀𝑥 ∈ ℋ (∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0))
3 ss2rab 4026 . . . . 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 236 . . . 4 (𝐴𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
54adantl 484 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
6 ocval 29042 . . . 4 (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
76ad2antlr 725 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐵 (𝑥 ·ih 𝑦) = 0})
8 ocval 29042 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
98ad2antrr 724 . . 3 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
105, 7, 93sstr4d 3993 . 2 (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
1110ex 415 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wral 3125  {crab 3129  wss 3913  cfv 6331  (class class class)co 7133  0cc0 10515  chba 28681   ·ih csp 28684  cort 28692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-hilex 28761
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-oc 29014
This theorem is referenced by:  occon2  29050  occon3  29059  ococin  29170  ssjo  29209  chsscon3i  29223  shjshsi  29254
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