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Mirrors > Home > HSE Home > Th. List > ocval | Structured version Visualization version GIF version |
Description: Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ocval | ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 27856 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | elpw2 4828 | . 2 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ) |
3 | raleq 3138 | . . . 4 ⊢ (𝑧 = 𝐻 → (∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0)) | |
4 | 3 | rabbidv 3189 | . . 3 ⊢ (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0}) |
5 | df-oc 28109 | . . 3 ⊢ ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0}) | |
6 | 1 | rabex 4813 | . . 3 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V |
7 | 4, 5, 6 | fvmpt 6282 | . 2 ⊢ (𝐻 ∈ 𝒫 ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0}) |
8 | 2, 7 | sylbir 225 | 1 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 𝒫 cpw 4158 ‘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: ocel 28140 ocsh 28142 occon 28146 chocvali 28158 |
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