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Theorem ocval 27985
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 27702 . . 3 ℋ ∈ V
21elpw2 4788 . 2 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 raleq 3127 . . . 4 (𝑧 = 𝐻 → (∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0))
43rabbidv 3177 . . 3 (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
5 df-oc 27955 . . 3 ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0})
61rabex 4773 . . 3 {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
74, 5, 6fvmpt 6239 . 2 (𝐻 ∈ 𝒫 ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
82, 7sylbir 225 1 (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  {crab 2911  wss 3555  𝒫 cpw 4130  cfv 5847  (class class class)co 6604  0cc0 9880  chil 27622   ·ih csp 27625  cort 27633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-hilex 27702
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-oc 27955
This theorem is referenced by:  ocel  27986  ocsh  27988  occon  27992  chocvali  28004
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