Theorem ocvval 21556

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 6898	. . 3 ⊢ 𝑉 ∈ V
3	2	elpw2 5338	. 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 21555	. . . . 5 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
9	8	fveq1d 6886	. . . 4 ⊢ (𝑊 ∈ V → ( ⊥ ‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10		raleq 3316	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 ))
11	10	rabbidv 3434	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
12		eqid 2726	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
13	2	rabex 5325	. . . . 5 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ∈ V
14	11, 12, 13	fvmpt 6991	. . . 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 2786	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
16		0fv 6928	. . . . 5 ⊢ (∅‘𝑆) = ∅
17		fvprc 6876	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
18	7, 17	eqtrid 2778	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
19	18	fveq1d 6886	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
20		ssrab2 4072	. . . . . 6 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21		fvprc 6876	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
22	1, 21	eqtrid 2778	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
23		sseq0 4394	. . . . . 6 ⊢ (({𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉 ∧ 𝑉 = ∅) → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
24	20, 22, 23	sylancr 586	. . . . 5 ⊢ (¬ 𝑊 ∈ V → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
25	16, 19, 24	3eqtr4a 2792	. . . 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 809	. 2 ⊢ (𝑆 ∈ 𝒫 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
28	3, 27	sylbir 234	1 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3055  {crab 3426  Vcvv 3468   ⊆ wss 3943  ∅c0 4317  𝒫 cpw 4597   ↦ cmpt 5224  ‘cfv 6536  (class class class)co 7404  Basecbs 17151  Scalarcsca 17207  ·_𝑖cip 17209  0_gc0g 17392  ocvcocv 21549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-ocv 21552

This theorem is referenced by:  elocv  21557  ocv0  21566  csscld  25128  hlhilocv  41343