Theorem isobs 21267

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 21252	. . . 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 7005	. . 3 ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
3		fveq2 6889	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		isobs.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2791	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4619	. . . . . . 7 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6889	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		isobs.h	. . . . . . . . . . . 12 ⊢ , = (·𝑖‘𝑊)
9	7, 8	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7423	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6889	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		isobs.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	eqtr4di 2791	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6893	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑊 → (1r‘(Scalar‘ℎ)) = (1r‘𝐹))
15		isobs.u	. . . . . . . . . . . 12 ⊢ 1 = (1r‘𝐹)
16	14, 15	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (1r‘(Scalar‘ℎ)) = 1 )
17	13	fveq2d 6893	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
18		isobs.z	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐹)
19	17, 18	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
20	16, 19	ifeq12d 4549	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) = if(𝑥 = 𝑦, 1 , 0 ))
21	10, 20	eqeq12d 2749	. . . . . . . . 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 3219	. . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = (ocv‘𝑊))
24		isobs.o	. . . . . . . . . . 11 ⊢ ⊥ = (ocv‘𝑊)
25	23, 24	eqtr4di 2791	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (ocv‘ℎ) = ⊥ )
26	25	fveq1d 6891	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → ((ocv‘ℎ)‘𝑏) = ( ⊥ ‘𝑏))
27		fveq2 6889	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (0g‘ℎ) = (0g‘𝑊))
28		isobs.y	. . . . . . . . . . 11 ⊢ 𝑌 = (0g‘𝑊)
29	27, 28	eqtr4di 2791	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (0g‘ℎ) = 𝑌)
30	29	sneqd 4640	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → {(0g‘ℎ)} = {𝑌})
31	26, 30	eqeq12d 2749	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)} ↔ ( ⊥ ‘𝑏) = {𝑌}))
32	22, 31	anbi12d 632	. . . . . . 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 3450	. . . . . 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 6903	. . . . . . . 8 ⊢ 𝑉 ∈ V
35	34	pwex 5378	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
36	35	rabex 5332	. . . . . 6 ⊢ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})} ∈ V
37	33, 1, 36	fvmpt 6996	. . . . 5 ⊢ (𝑊 ∈ PreHil → (OBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝑏) = {𝑌})})
38	37	eleq2d 2820	. . . 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 3323	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
40	39	raleqbi1dv 3334	. . . . . . 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 6898	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (( ⊥ ‘𝑏) = {𝑌} ↔ ( ⊥ ‘𝐵) = {𝑌}))
42	40, 41	anbi12d 632	. . . . . 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 3683	. . . . 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 5345	. . . . . 6 ⊢ (𝐵 ∈ 𝒫 𝑉 ↔ 𝐵 ⊆ 𝑉)
45	44	anbi1i 625	. . . . 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 275	. . . 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 287	. . 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 821	. 2 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ (𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌}))))
49		3anass 1096	. 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 278	1 ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433   ⊆ wss 3948  ifcif 4528  𝒫 cpw 4602  {csn 4628  ‘cfv 6541  (class class class)co 7406  Basecbs 17141  Scalarcsca 17197  ·_𝑖cip 17199  0_gc0g 17382  1_rcur 19999  PreHilcphl 21169  ocvcocv 21205  OBasiscobs 21249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fv 6549  df-ov 7409  df-obs 21252

This theorem is referenced by:  obsip  21268  obsrcl  21270  obsss  21271  obsocv  21273