Theorem ocvval 21597

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	1	fvexi 6831	. . 3 ⊢ 𝑉 ∈ V
3	2	elpw2 5270	. 2 ⊢ (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 ⊆ 𝑉)
4		ocvfval.i	. . . . . 6 ⊢ , = (·𝑖‘𝑊)
5		ocvfval.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
6		ocvfval.z	. . . . . 6 ⊢ 0 = (0g‘𝐹)
7		ocvfval.o	. . . . . 6 ⊢ ⊥ = (ocv‘𝑊)
8	1, 4, 5, 6, 7	ocvfval 21596	. . . . 5 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
9	8	fveq1d 6819	. . . 4 ⊢ (𝑊 ∈ V → ( ⊥ ‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10		raleq 3287	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 ))
11	10	rabbidv 3400	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
12		eqid 2730	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
13	2	rabex 5275	. . . . 5 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ∈ V
14	11, 12, 13	fvmpt 6924	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
15	9, 14	sylan9eq 2785	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
16		0fv 6858	. . . . 5 ⊢ (∅‘𝑆) = ∅
17		fvprc 6809	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
18	7, 17	eqtrid 2777	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
19	18	fveq1d 6819	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
20		ssrab2 4028	. . . . . 6 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21		fvprc 6809	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
22	1, 21	eqtrid 2777	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
23		sseq0 4351	. . . . . 6 ⊢ (({𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉 ∧ 𝑉 = ∅) → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
24	20, 22, 23	sylancr 587	. . . . 5 ⊢ (¬ 𝑊 ∈ V → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
25	16, 19, 24	3eqtr4a 2791	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
26	25	adantr 480	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
27	15, 26	pm2.61ian 811	. 2 ⊢ (𝑆 ∈ 𝒫 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
28	3, 27	sylbir 235	1 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2110  ∀wral 3045  {crab 3393  Vcvv 3434   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4548   ↦ cmpt 5170  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  Scalarcsca 17156  ·_𝑖cip 17158  0_gc0g 17335  ocvcocv 21590

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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-fv 6485  df-ov 7344  df-ocv 21593

This theorem is referenced by:  elocv  21598  ocv0  21607  csscld  25169  hlhilocv  41975