Theorem ocvval 21220

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 6906	. . 3 ⊢ 𝑉 ∈ V
3	2	elpw2 5346	. 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 21219	. . . . 5 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
9	8	fveq1d 6894	. . . 4 ⊢ (𝑊 ∈ V → ( ⊥ ‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10		raleq 3323	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 ))
11	10	rabbidv 3441	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
12		eqid 2733	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
13	2	rabex 5333	. . . . 5 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ∈ V
14	11, 12, 13	fvmpt 6999	. . . 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 2793	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
16		0fv 6936	. . . . 5 ⊢ (∅‘𝑆) = ∅
17		fvprc 6884	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
18	7, 17	eqtrid 2785	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
19	18	fveq1d 6894	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
20		ssrab2 4078	. . . . . 6 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21		fvprc 6884	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
22	1, 21	eqtrid 2785	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
23		sseq0 4400	. . . . . 6 ⊢ (({𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉 ∧ 𝑉 = ∅) → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
24	20, 22, 23	sylancr 588	. . . . 5 ⊢ (¬ 𝑊 ∈ V → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } = ∅)
25	16, 19, 24	3eqtr4a 2799	. . . 4 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
26	25	adantr 482	. . 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 234	1 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603   ↦ cmpt 5232  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200  ·_𝑖cip 17202  0_gc0g 17385  ocvcocv 21213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-ocv 21216

This theorem is referenced by:  elocv  21221  ocv0  21230  csscld  24766  hlhilocv  40832