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Theorem ocval 28838
Description: Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ocval (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
Distinct variable group:   𝑥,𝑦,𝐻

Proof of Theorem ocval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28555 . . 3 ℋ ∈ V
21elpw2 5104 . 2 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 raleq 3345 . . . 4 (𝑧 = 𝐻 → (∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0))
43rabbidv 3403 . . 3 (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
5 df-oc 28808 . . 3 ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0})
61rabex 5091 . . 3 {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
74, 5, 6fvmpt 6595 . 2 (𝐻 ∈ 𝒫 ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
82, 7sylbir 227 1 (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  wral 3088  {crab 3092  wss 3829  𝒫 cpw 4422  cfv 6188  (class class class)co 6976  0cc0 10335  chba 28475   ·ih csp 28478  cort 28486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-hilex 28555
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-oc 28808
This theorem is referenced by:  ocel  28839  ocsh  28841  occon  28845  chocvali  28857
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