Theorem ocval 31429

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 31148	. . 3 ⊢ ℋ ∈ V
2	1	elpw2 5289	. 2 ⊢ (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3		raleq 3316	. . . 4 ⊢ (𝑧 = 𝐻 → (∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0 ↔ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0))
4	3	rabbidv 3420	. . 3 ⊢ (𝑧 = 𝐻 → {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0} = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0})
5		df-oc 31401	. . 3 ⊢ ⊥ = (𝑧 ∈ 𝒫 ℋ ↦ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝑧 (𝑥 ·ih 𝑦) = 0})
6	1	rabex 5294	. . 3 ⊢ {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0} ∈ V
7	4, 5, 6	fvmpt 6971	. 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 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554  ‘cfv 6517  (class class class)co 7392  0cc0 11070   ℋchba 31068   ·_ih csp 31071  ⊥cort 31079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-hilex 31148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-oc 31401

This theorem is referenced by:  ocel  31430  ocsh  31432  occon  31436  chocvali  31448