Theorem chocvali 31318

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	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
chocvali	⊢ (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐴 (𝑥 ·ih 𝑦) = 0}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem chocvali

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}

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   C_ℋ 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)