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Theorem ocval 28473
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 28190 . . 3 ℋ ∈ V
21elpw2 4956 . 2 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 raleq 3286 . . . 4 (𝑧 = 𝐻 → (∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0))
43rabbidv 3338 . . 3 (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})
5 df-oc 28443 . . 3 ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦𝑧 (𝑥 ·ih 𝑦) = 0})
61rabex 4943 . . 3 {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
74, 5, 6fvmpt 6424 . 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 1630  wcel 2144  wral 3060  {crab 3064  wss 3721  𝒫 cpw 4295  cfv 6031  (class class class)co 6792  0cc0 10137  chil 28110   ·ih csp 28113  cort 28121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-hilex 28190
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-oc 28443
This theorem is referenced by:  ocel  28474  ocsh  28476  occon  28480  chocvali  28492
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