Theorem chocvali 28158

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 28088	. 2 ⊢ 𝐴 ⊆ ℋ
3		ocval 28139	. 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 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  0cc0 9936   ℋchil 27776   ·_ih csp 27779   C_ℋ cch 27786  ⊥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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-sh 28064  df-ch 28078  df-oc 28109

This theorem is referenced by: (None)