Theorem ocval 29059

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 ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})

Distinct variable group:   𝑥,𝑦,𝐻

Proof of Theorem ocval

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 28778	. . 3 ⊢ ℋ ∈ V
2	1	elpw2 5250	. 2 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		raleq 3407	. . . 4 ⊢ (𝑧 = 𝐻 → (∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0))
4	3	rabbidv 3482	. . 3 ⊢ (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})
5		df-oc 29031	. . 3 ⊢ ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0})
6	1	rabex 5237	. . 3 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
7	4, 5, 6	fvmpt 6770	. 2 ⊢ (𝐻 ∈ 𝒫 ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})
8	2, 7	sylbir 237	1 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144   ⊆ wss 3938  𝒫 cpw 4541  ‘cfv 6357  (class class class)co 7158  0cc0 10539   ℋchba 28698   ·_ih csp 28701  ⊥cort 28709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-oc 29031

This theorem is referenced by:  ocel  29060  ocsh  29062  occon  29066  chocvali  29078