Theorem occon 29070

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 3981	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
2	1	ralrimivw 3150	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ ℋ (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
3		ss2rab 3998	. . . . 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 237	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
5	4	adantl 485	. . 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 29063	. . . 4 ⊢ (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
7	6	ad2antlr 726	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
8		ocval 29063	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
9	8	ad2antrr 725	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
10	5, 7, 9	3sstr4d 3962	. 2 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
11	10	ex 416	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∀wral 3106  {crab 3110   ⊆ wss 3881  ‘cfv 6324  (class class class)co 7135  0cc0 10526   ℋchba 28702   ·_ih csp 28705  ⊥cort 28713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28782

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-oc 29035

This theorem is referenced by:  occon2  29071  occon3  29080  ococin  29191  ssjo  29230  chsscon3i  29244  shjshsi  29275