HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chocvali Structured version   Visualization version   GIF version

Theorem chocvali 30188
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.)
Hypothesis
Ref Expression
chocval.1 𝐴C
Assertion
Ref Expression
chocvali (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem chocvali
StepHypRef Expression
1 chocval.1 . . 3 𝐴C
21chssii 30120 . 2 𝐴 ⊆ ℋ
3 ocval 30169 . 2 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
42, 3ax-mp 5 1 (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wss 3910  cfv 6496  (class class class)co 7356  0cc0 11050  chba 29808   ·ih csp 29811   C cch 29818  cort 29819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-hilex 29888
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7359  df-sh 30096  df-ch 30110  df-oc 30141
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator