Theorem elocv 21577

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 6895	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ dom ⊥ )
2		n0i 4303	. . . . . . . . 9 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ¬ ( ⊥ ‘𝑆) = ∅)
3		ocvfval.o	. . . . . . . . . . . 12 ⊢ ⊥ = (ocv‘𝑊)
4		fvprc 6850	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
5	3, 4	eqtrid 2776	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
6	5	fveq1d 6860	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
7		0fv 6902	. . . . . . . . . 10 ⊢ (∅‘𝑆) = ∅
8	6, 7	eqtrdi 2780	. . . . . . . . 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 21575	. . . . . . . 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 5869	. . . . . 6 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
17	10	fvexi 6872	. . . . . . . 8 ⊢ 𝑉 ∈ V
18	17	rabex 5294	. . . . . . 7 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19		eqid 2729	. . . . . . 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 6662	. . . . . 6 ⊢ dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
21	16, 20	eqtrdi 2780	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = 𝒫 𝑉)
22	1, 21	eleqtrd 2830	. . . 4 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ 𝒫 𝑉)
23	22	elpwid 4572	. . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ⊆ 𝑉)
24	10, 11, 12, 13, 3	ocvval 21576	. . . . 5 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 })
25	24	eleq2d 2814	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ 𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 }))
26		oveq1 7394	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
27	26	eqeq1d 2731	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
28	27	ralbidv 3156	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
29	28	elrab 3659	. . . 4 ⊢ (𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
30	25, 29	bitrdi 287	. . 3 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
31	23, 30	biadanii 821	. 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
32		3anass 1094	. 2 ⊢ ((𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
33	31, 32	bitr4i 278	1 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563   ↦ cmpt 5188  dom cdm 5638  ‘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:  ocvi  21578  ocvss  21579  ocvocv  21580  ocvlss  21581  ocv2ss  21582  unocv  21589  iunocv  21590  obselocv  21637  clsocv  25150  pjthlem2  25338