|   | Hilbert Space Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > HSE Home > Th. List > chocvali | Structured version Visualization version GIF version | ||
| Description: Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of 𝐴 is the set of vectors that are orthogonal to all vectors in 𝐴. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| chocval.1 | ⊢ 𝐴 ∈ Cℋ | 
| Ref | Expression | 
|---|---|
| chocvali | ⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | chocval.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
| 2 | 1 | chssii 31250 | . 2 ⊢ 𝐴 ⊆ ℋ | 
| 3 | ocval 31299 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}) | |
| 4 | 2, 3 | ax-mp 5 | 1 ⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℋchba 30938 ·ih csp 30941 Cℋ cch 30948 ⊥cort 30949 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-sh 31226 df-ch 31240 df-oc 31271 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |