Theorem isobs 22006

Description: The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
isobs.v	⊢ 𝑉 = (Base‘𝑊)
isobs.h	⊢ , = (·𝑖‘𝑊)
isobs.f	⊢ 𝐹 = (Scalar‘𝑊)
isobs.u	⊢ 1 = (1r‘𝐹)
isobs.z	⊢ 0 = (0g‘𝐹)
isobs.o	⊢ ⊥ = (ocv‘𝑊)
isobs.y	⊢ 𝑌 = (0g‘𝑊)

Assertion

Ref	Expression
isobs	⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))

Distinct variable groups:   𝑥,𝑦, ,   𝑥, 0 ,𝑦   𝑥, 1 ,𝑦   𝑥,𝐵,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝐹(𝑥, 𝑦)   ⊥ (𝑥, 𝑦)   𝑉(𝑥, 𝑦)   𝑌(𝑥, 𝑦)

Proof of Theorem isobs

Dummy variables ℎ 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-obs 21991	. . . 4 ⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})
2	1	mptrcl 6995	. . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
3		fveq2 6877	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		isobs.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2814	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4574	. . . . . . 7 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6877	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		isobs.h	. . . . . . . . . . . 12 ⊢ , = (·𝑖‘𝑊)
9	7, 8	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7429	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6877	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		isobs.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	eqtr4di 2814	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6881	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑊 → (1r‘(Scalar‘ℎ)) = (1r‘𝐹))
15		isobs.u	. . . . . . . . . . . 12 ⊢ 1 = (1r‘𝐹)
16	14, 15	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (1r‘(Scalar‘ℎ)) = 1 )
17	13	fveq2d 6881	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
18		isobs.z	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐹)
19	17, 18	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
20	16, 19	ifeq12d 4504	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) = if(𝑥 = 𝑦, 1 , 0 ))
21	10, 20	eqeq12d 2777	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → ((𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ↔ (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
22	21	2ralbidv 3227	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ↔ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
23		fveq2 6877	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
24		isobs.o	. . . . . . . . . . 11 ⊢ ⊥ = (ocv‘𝑊)
25	23, 24	eqtr4di 2814	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
26	25	fveq1d 6879	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → ((ocv‘ℎ)‘𝑏) = ( ⊥ ‘𝑏))
27		fveq2 6877	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (0g‘ℎ) = (0g‘𝑊))
28		isobs.y	. . . . . . . . . . 11 ⊢ 𝑌 = (0g‘𝑊)
29	27, 28	eqtr4di 2814	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (0g‘ℎ) = 𝑌)
30	29	sneqd 4596	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → {(0g‘ℎ)} = {𝑌})
31	26, 30	eqeq12d 2777	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)} ↔ ( ⊥ ‘𝑏) = {𝑌}))
32	22, 31	anbi12d 644	. . . . . . 7 ⊢ (ℎ = 𝑊 → ((∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)}) ↔ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})))
33	6, 32	rabeqbidv 3430	. . . . . 6 ⊢ (ℎ = 𝑊 → {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})} = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})})
34	4	fvexi 6891	. . . . . . . 8 ⊢ 𝑉 ∈ V
35	34	pwex 5342	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
36	35	rabex 5300	. . . . . 6 ⊢ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})} ∈ V
37	33, 1, 36	fvmpt 6985	. . . . 5 ⊢ (𝑊 ∈ PreHil → (OBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})})
38	37	eleq2d 2847	. . . 4 ⊢ (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})}))
39		raleq 3317	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
40	39	raleqbi1dv 3330	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
41		fveqeq2 6886	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (( ⊥ ‘𝑏) = {𝑌} ↔ ( ⊥ ‘𝐵) = {𝑌}))
42	40, 41	anbi12d 644	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌}) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))
43	42	elrab 3645	. . . . 5 ⊢ (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})} ↔ (𝐵 ∈ 𝒫 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))
44	34	elpw2 5296	. . . . . 6 ⊢ (𝐵 ∈ 𝒫 𝑉 ↔ 𝐵 ⊆ 𝑉)
45	44	anbi1i 636	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})) ↔ (𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))
46	43, 45	bitri 278	. . . 4 ⊢ (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})} ↔ (𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))
47	38, 46	bitrdi 290	. . 3 ⊢ (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ (𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌}))))
48	2, 47	biadanii 834	. 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ (𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌}))))
49		3anass 1111	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})) ↔ (𝑊 ∈ PreHil ∧ (𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌}))))
50	48, 49	bitr4i 281	1 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899  ifcif 4482  𝒫 cpw 4557  {csn 4584  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  Scalarcsca 17411  ·_𝑖cip 17413  0_gc0g 17590  1_rcur 20387  PreHilcphl 21910  ocvcocv 21946  OBasiscobs 21988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-obs 21991

This theorem is used by:  obsip  22007  obsrcl  22009  obsss  22010  obsocv  22012