Theorem elocv 20812

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 6702	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ dom ⊥ )
2		n0i 4299	. . . . . . . . 9 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ¬ ( ⊥ ‘𝑆) = ∅)
3		ocvfval.o	. . . . . . . . . . . 12 ⊢ ⊥ = (ocv‘𝑊)
4		fvprc 6663	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
5	3, 4	syl5eq 2868	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
6	5	fveq1d 6672	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
7		0fv 6709	. . . . . . . . . 10 ⊢ (∅‘𝑆) = ∅
8	6, 7	syl6eq 2872	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = ∅)
9	2, 8	nsyl2 143	. . . . . . . 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 20810	. . . . . . . 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 5774	. . . . . 6 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
17	10	fvexi 6684	. . . . . . . 8 ⊢ 𝑉 ∈ V
18	17	rabex 5235	. . . . . . 7 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19		eqid 2821	. . . . . . 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 6492	. . . . . 6 ⊢ dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
21	16, 20	syl6eq 2872	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = 𝒫 𝑉)
22	1, 21	eleqtrd 2915	. . . 4 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ 𝒫 𝑉)
23	22	elpwid 4550	. . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ⊆ 𝑉)
24	10, 11, 12, 13, 3	ocvval 20811	. . . . 5 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 })
25	24	eleq2d 2898	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ 𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 }))
26		oveq1 7163	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
27	26	eqeq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
28	27	ralbidv 3197	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
29	28	elrab 3680	. . . 4 ⊢ (𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
30	25, 29	syl6bb 289	. . 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 1091	. 2 ⊢ ((𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
33	31, 32	bitr4i 280	1 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539   ↦ cmpt 5146  dom cdm 5555  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Scalarcsca 16568  ·_𝑖cip 16570  0_gc0g 16713  ocvcocv 20804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-ocv 20807

This theorem is referenced by:  ocvi  20813  ocvss  20814  ocvocv  20815  ocvlss  20816  ocv2ss  20817  unocv  20824  iunocv  20825  obselocv  20872  clsocv  23853  pjthlem2  24041