Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > occon | Structured version Visualization version GIF version |
Description: Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
occon | ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssralv 3666 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0)) | |
2 | 1 | ralrimivw 2967 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ ℋ (∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0 → ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0)) |
3 | ss2rab 3678 | . . . . 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 224 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0} ⊆ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}) |
5 | 4 | adantl 482 | . . 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 28139 | . . . 4 ⊢ (𝐵 ⊆ ℋ → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0}) | |
7 | 6 | ad2antlr 763 | . . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐵 (𝑥 ·ih 𝑦) = 0}) |
8 | ocval 28139 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}) | |
9 | 8 | ad2antrr 762 | . . 3 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}) |
10 | 5, 7, 9 | 3sstr4d 3648 | . 2 ⊢ (((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘𝐵) ⊆ (⊥‘𝐴)) |
11 | 10 | ex 450 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∀wral 2912 {crab 2916 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℋchil 27776 ·ih csp 27779 ⊥cort 27787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-oc 28109 |
This theorem is referenced by: occon2 28147 occon3 28156 ococin 28267 ssjo 28306 chsscon3i 28320 shjshsi 28351 |
Copyright terms: Public domain | W3C validator |