Theorem elocv 21072

Description: Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed 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
elocv	⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))

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

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

Proof of Theorem elocv

Dummy variables 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6879	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ dom ⊥ )
2		n0i 4293	. . . . . . . . 9 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ¬ ( ⊥ ‘𝑆) = ∅)
3		ocvfval.o	. . . . . . . . . . . 12 ⊢ ⊥ = (ocv‘𝑊)
4		fvprc 6834	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
5	3, 4	eqtrid 2788	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
6	5	fveq1d 6844	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
7		0fv 6886	. . . . . . . . . 10 ⊢ (∅‘𝑆) = ∅
8	6, 7	eqtrdi 2792	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = ∅)
9	2, 8	nsyl2 141	. . . . . . . 8 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑊 ∈ V)
10		ocvfval.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
11		ocvfval.i	. . . . . . . . 9 ⊢ , = (·𝑖‘𝑊)
12		ocvfval.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
13		ocvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐹)
14	10, 11, 12, 13, 3	ocvfval 21070	. . . . . . . 8 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
15	9, 14	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
16	15	dmeqd 5861	. . . . . 6 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
17	10	fvexi 6856	. . . . . . . 8 ⊢ 𝑉 ∈ V
18	17	rabex 5289	. . . . . . 7 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19		eqid 2736	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 })
20	18, 19	dmmpti 6645	. . . . . 6 ⊢ dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
21	16, 20	eqtrdi 2792	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = 𝒫 𝑉)
22	1, 21	eleqtrd 2840	. . . 4 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ 𝒫 𝑉)
23	22	elpwid 4569	. . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ⊆ 𝑉)
24	10, 11, 12, 13, 3	ocvval 21071	. . . . 5 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 })
25	24	eleq2d 2823	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ 𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 }))
26		oveq1 7364	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
27	26	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
28	27	ralbidv 3174	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
29	28	elrab 3645	. . . 4 ⊢ (𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
30	25, 29	bitrdi 286	. . 3 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
31	23, 30	biadanii 820	. 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
32		3anass 1095	. 2 ⊢ ((𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
33	31, 32	bitr4i 277	1 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560   ↦ cmpt 5188  dom cdm 5633  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  Scalarcsca 17136  ·_𝑖cip 17138  0_gc0g 17321  ocvcocv 21064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-ocv 21067

This theorem is referenced by:  ocvi  21073  ocvss  21074  ocvocv  21075  ocvlss  21076  ocv2ss  21077  unocv  21084  iunocv  21085  obselocv  21134  clsocv  24614  pjthlem2  24802