Theorem elocv 21980

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 6920	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ dom ⊥ )
2		n0i 4285	. . . . . . . . 9 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ¬ ( ⊥ ‘𝑆) = ∅)
3		ocvfval.o	. . . . . . . . . . . 12 ⊢ ⊥ = (ocv‘𝑊)
4		fvprc 6878	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (ocv‘𝑊) = ∅)
5	3, 4	eqtrid 2807	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → ⊥ = ∅)
6	5	fveq1d 6888	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = (∅‘𝑆))
7		0fv 6927	. . . . . . . . . 10 ⊢ (∅‘𝑆) = ∅
8	6, 7	eqtrdi 2811	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → ( ⊥ ‘𝑆) = ∅)
9	2, 8	nsyl2 142	. . . . . . . 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 21978	. . . . . . . 8 ⊢ (𝑊 ∈ V → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
15	9, 14	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
16	15	dmeqd 5886	. . . . . 6 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }))
17	10	fvexi 6900	. . . . . . . 8 ⊢ 𝑉 ∈ V
18	17	rabex 5299	. . . . . . 7 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19		eqid 2760	. . . . . . 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 6684	. . . . . 6 ⊢ dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
21	16, 20	eqtrdi 2811	. . . . 5 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → dom ⊥ = 𝒫 𝑉)
22	1, 21	eleqtrd 2862	. . . 4 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ∈ 𝒫 𝑉)
23	22	elpwid 4565	. . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → 𝑆 ⊆ 𝑉)
24	10, 11, 12, 13, 3	ocvval 21979	. . . . 5 ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 })
25	24	eleq2d 2846	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ 𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 }))
26		oveq1 7428	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
27	26	eqeq1d 2762	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
28	27	ralbidv 3185	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
29	28	elrab 3644	. . . 4 ⊢ (𝐴 ∈ {𝑦 ∈ 𝑉 ∣ ∀𝑥 ∈ 𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
30	25, 29	bitrdi 290	. . 3 ⊢ (𝑆 ⊆ 𝑉 → (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
31	23, 30	biadanii 834	. 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
32		3anass 1111	. 2 ⊢ ((𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆 ⊆ 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )))
33	31, 32	bitr4i 281	1 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556   ↦ cmpt 5185  dom cdm 5650  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  Scalarcsca 17437  ·_𝑖cip 17439  0_gc0g 17616  ocvcocv 21972

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-ocv 21975

This theorem is used by:  ocvi  21981  ocvss  21982  ocvocv  21983  ocvlss  21984  ocv2ss  21985  unocv  21992  iunocv  21993  obselocv  22040  clsocv  25577  pjthlem2  25765