Theorem occon 31249

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 4006	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
2	1	adantr 480	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
3	2	ss2rabdv 4029	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
4	3	adantl 481	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
5		ocval 31242	. . . 4 ⊢ (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
6	5	ad2antlr 727	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
7		ocval 31242	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
8	7	ad2antrr 726	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
9	4, 6, 8	3sstr4d 3993	. 2 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
10	9	ex 412	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   ⊆ wss 3905  ‘cfv 6486  (class class class)co 7353  0cc0 11028   ℋchba 30881   ·_ih csp 30884  ⊥cort 30892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-hilex 30961

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-oc 31214

This theorem is referenced by:  occon2  31250  occon3  31259  ococin  31370  ssjo  31409  chsscon3i  31423  shjshsi  31454