Theorem ocvval 21576

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 6872	. . 3 ⊢ 𝑉 ∈ V
3	2	elpw2 5289	. 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 21575	. . . . 5 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
9	8	fveq1d 6860	. . . 4 ⊢ (𝑊 ∈ V → ( ⊥ ‘𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10		raleq 3296	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 ))
11	10	rabbidv 3413	. . . . 5 ⊢ (𝑠 = 𝑆 → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
12		eqid 2729	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
13	2	rabex 5294	. . . . 5 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ∈ V
14	11, 12, 13	fvmpt 6968	. . . 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 2784	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 })
16		0fv 6902	. . . . 5 ⊢ (∅‘𝑆) = ∅
17		fvprc 6850	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
18	7, 17	eqtrid 2776	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
19	18	fveq1d 6860	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
20		ssrab2 4043	. . . . . 6 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21		fvprc 6850	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
22	1, 21	eqtrid 2776	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
23		sseq0 4366	. . . . . 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 2790	. . . 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 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Scalarcsca 17223  ·_𝑖cip 17225  0_gc0g 17402  ocvcocv 21569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-ocv 21572

This theorem is referenced by:  elocv  21577  ocv0  21586  csscld  25149  hlhilocv  41951