Theorem occon 31436

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 4005	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
2	1	adantr 484	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0))
3	2	ss2rabdv 4028	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
4	3	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})
5		ocval 31429	. . . 4 ⊢ (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
6	5	ad2antlr 737	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0})
7		ocval 31429	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
8	7	ad2antrr 736	. . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0})
9	4, 6, 8	3sstr4d 3991	. 2 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴))
10	9	ex 416	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3904  ‘cfv 6517  (class class class)co 7392  0cc0 11070   ℋchba 31068   ·_ih csp 31071  ⊥cort 31079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-hilex 31148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-oc 31401

This theorem is referenced by:  occon2  31437  occon3  31446  ococin  31557  ssjo  31596  chsscon3i  31610  shjshsi  31641