Theorem ocvval 19830

Description: Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
ocvfval.v	⊢ 𝑉 = (Base‘𝑊)
ocvfval.i	⊢ , = (·𝑖‘𝑊)
ocvfval.f	⊢ 𝐹 = (Scalar‘𝑊)
ocvfval.z	⊢ 0 = (0g‘𝐹)
ocvfval.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
ocvval	⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })

Distinct variable groups:   𝑥,𝑦, 0   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥, , ,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)   ⊥ (𝑥,𝑦)

Proof of Theorem ocvval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocvfval.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		fvex 6113	. . . 4 ⊢ (Base‘𝑊) ∈ V
3	1, 2	eqeltri 2684	. . 3 ⊢ 𝑉 ∈ V
4	3	elpw2 4755	. 2 ⊢ (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉)
5		ocvfval.i	. . . . . 6 ⊢ , = (·𝑖‘𝑊)
6		ocvfval.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
7		ocvfval.z	. . . . . 6 ⊢ 0 = (0g‘𝐹)
8		ocvfval.o	. . . . . 6 ⊢ ⊥ = (ocv‘𝑊)
9	1, 5, 6, 7, 8	ocvfval 19829	. . . . 5 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
10	9	fveq1d 6105	. . . 4 ⊢ (𝑊 ∈ V → ( ⊥ ‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
11		raleq 3115	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 ))
12	11	rabbidv 3164	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
13		eqid 2610	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
14	3	rabex 4740	. . . . 5 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ∈ V
15	12, 13, 14	fvmpt 6191	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
16	10, 15	sylan9eq 2664	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
17		0fv 6137	. . . . 5 ⊢ (∅‘𝑆) = ∅
18		fvprc 6097	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
19	8, 18	syl5eq 2656	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
20	19	fveq1d 6105	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
21		ssrab2 3650	. . . . . 6 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
22		fvprc 6097	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
23	1, 22	syl5eq 2656	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
24		sseq0 3927	. . . . . 6 ⊢ (({𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉 ∧ 𝑉 = ∅) → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
25	21, 23, 24	sylancr 694	. . . . 5 ⊢ (¬ 𝑊 ∈ V → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
26	17, 20, 25	3eqtr4a 2670	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
27	26	adantr 480	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
28	16, 27	pm2.61ian 827	. 2 ⊢ (𝑆 ∈ 𝒫 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
29	4, 28	sylbir 224	1 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771  ·_𝑖cip 15773  0_gc0g 15923  ocvcocv 19823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-ocv 19826

This theorem is referenced by:  elocv  19831  ocv0  19840  csscld  22856  hlhilocv  36267