Theorem chocvali 29082

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 29014	. 2 ⊢ 𝐴 ⊆ ℋ
3		ocval 29063	. 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 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ⊆ wss 3881  ‘cfv 6324  (class class class)co 7135  0cc0 10526   ℋchba 28702   ·_ih csp 28705   C_ℋ cch 28712  ⊥cort 28713

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28782

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-sh 28990  df-ch 29004  df-oc 29035

This theorem is referenced by: (None)