Theorem ocval 31669

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 31388	. . 3 ⊢ ℋ ∈ V
2	1	elpw2 5310	. 2 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		raleq 3323	. . . 4 ⊢ (𝑧 = 𝐻 → (∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0))
4	3	rabbidv 3426	. . 3 ⊢ (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})
5		df-oc 31641	. . 3 ⊢ ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0})
6	1	rabex 5314	. . 3 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
7	4, 5, 6	fvmpt 6996	. 2 ⊢ (𝐻 ∈ 𝒫 ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})
8	2, 7	sylbir 238	1 ⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419   ⊆ wss 3908  𝒫 cpw 4567  ‘cfv 6543  (class class class)co 7423  0cc0 11118   ℋchba 31308   ·_ih csp 31311  ⊥cort 31319

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-hilex 31388

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-oc 31641

This theorem is used by:  ocel  31670  ocsh  31672  occon  31676  chocvali  31688