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Theorem isobs 19883
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 , = (·𝑖𝑊)
isobs.f 𝐹 = (Scalar‘𝑊)
isobs.u 1 = (1r𝐹)
isobs.z 0 = (0g𝐹)
isobs.o = (ocv‘𝑊)
isobs.y 𝑌 = (0g𝑊)
Assertion
Ref Expression
isobs (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (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 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-obs 19868 . . . . 5 OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
21dmmptss 5548 . . . 4 dom OBasis ⊆ PreHil
3 elfvdm 6130 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ dom OBasis)
42, 3sseldi 3566 . . 3 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
5 fveq2 6103 . . . . . . . . 9 ( = 𝑊 → (Base‘) = (Base‘𝑊))
6 isobs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
75, 6syl6eqr 2662 . . . . . . . 8 ( = 𝑊 → (Base‘) = 𝑉)
87pweqd 4113 . . . . . . 7 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
9 fveq2 6103 . . . . . . . . . . . 12 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
10 isobs.h . . . . . . . . . . . 12 , = (·𝑖𝑊)
119, 10syl6eqr 2662 . . . . . . . . . . 11 ( = 𝑊 → (·𝑖) = , )
1211oveqd 6566 . . . . . . . . . 10 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
13 fveq2 6103 . . . . . . . . . . . . . 14 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
14 isobs.f . . . . . . . . . . . . . 14 𝐹 = (Scalar‘𝑊)
1513, 14syl6eqr 2662 . . . . . . . . . . . . 13 ( = 𝑊 → (Scalar‘) = 𝐹)
1615fveq2d 6107 . . . . . . . . . . . 12 ( = 𝑊 → (1r‘(Scalar‘)) = (1r𝐹))
17 isobs.u . . . . . . . . . . . 12 1 = (1r𝐹)
1816, 17syl6eqr 2662 . . . . . . . . . . 11 ( = 𝑊 → (1r‘(Scalar‘)) = 1 )
1915fveq2d 6107 . . . . . . . . . . . 12 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
20 isobs.z . . . . . . . . . . . 12 0 = (0g𝐹)
2119, 20syl6eqr 2662 . . . . . . . . . . 11 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
2218, 21ifeq12d 4056 . . . . . . . . . 10 ( = 𝑊 → if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) = if(𝑥 = 𝑦, 1 , 0 ))
2312, 22eqeq12d 2625 . . . . . . . . 9 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
24232ralbidv 2972 . . . . . . . 8 ( = 𝑊 → (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
25 fveq2 6103 . . . . . . . . . . 11 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
26 isobs.o . . . . . . . . . . 11 = (ocv‘𝑊)
2725, 26syl6eqr 2662 . . . . . . . . . 10 ( = 𝑊 → (ocv‘) = )
2827fveq1d 6105 . . . . . . . . 9 ( = 𝑊 → ((ocv‘)‘𝑏) = ( 𝑏))
29 fveq2 6103 . . . . . . . . . . 11 ( = 𝑊 → (0g) = (0g𝑊))
30 isobs.y . . . . . . . . . . 11 𝑌 = (0g𝑊)
3129, 30syl6eqr 2662 . . . . . . . . . 10 ( = 𝑊 → (0g) = 𝑌)
3231sneqd 4137 . . . . . . . . 9 ( = 𝑊 → {(0g)} = {𝑌})
3328, 32eqeq12d 2625 . . . . . . . 8 ( = 𝑊 → (((ocv‘)‘𝑏) = {(0g)} ↔ ( 𝑏) = {𝑌}))
3424, 33anbi12d 743 . . . . . . 7 ( = 𝑊 → ((∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)}) ↔ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})))
358, 34rabeqbidv 3168 . . . . . 6 ( = 𝑊 → {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})} = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})})
36 fvex 6113 . . . . . . . . 9 (Base‘𝑊) ∈ V
376, 36eqeltri 2684 . . . . . . . 8 𝑉 ∈ V
3837pwex 4774 . . . . . . 7 𝒫 𝑉 ∈ V
3938rabex 4740 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})} ∈ V
4035, 1, 39fvmpt 6191 . . . . 5 (𝑊 ∈ PreHil → (OBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})})
4140eleq2d 2673 . . . 4 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})}))
42 raleq 3115 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
4342raleqbi1dv 3123 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
44 fveq2 6103 . . . . . . . 8 (𝑏 = 𝐵 → ( 𝑏) = ( 𝐵))
4544eqeq1d 2612 . . . . . . 7 (𝑏 = 𝐵 → (( 𝑏) = {𝑌} ↔ ( 𝐵) = {𝑌}))
4643, 45anbi12d 743 . . . . . 6 (𝑏 = 𝐵 → ((∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌}) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
4746elrab 3331 . . . . 5 (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})} ↔ (𝐵 ∈ 𝒫 𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
4837elpw2 4755 . . . . . 6 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
4948anbi1i 727 . . . . 5 ((𝐵 ∈ 𝒫 𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})) ↔ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
5047, 49bitri 263 . . . 4 (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})} ↔ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
5141, 50syl6bb 275 . . 3 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
524, 51biadan2 672 . 2 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
53 3anass 1035 . 2 ((𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})) ↔ (𝑊 ∈ PreHil ∧ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
5452, 53bitr4i 266 1 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  wss 3540  ifcif 4036  𝒫 cpw 4108  {csn 4125  dom cdm 5038  cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771  ·𝑖cip 15773  0gc0g 15923  1rcur 18324  PreHilcphl 19788  ocvcocv 19823  OBasiscobs 19865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-obs 19868
This theorem is referenced by:  obsip  19884  obsrcl  19886  obsss  19887  obsocv  19889
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