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Theorem isobs 19983
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 19968 . . . . 5 OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
21dmmptss 5590 . . . 4 dom OBasis ⊆ PreHil
3 elfvdm 6177 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ dom OBasis)
42, 3sseldi 3581 . . 3 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
5 fveq2 6148 . . . . . . . . 9 ( = 𝑊 → (Base‘) = (Base‘𝑊))
6 isobs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
75, 6syl6eqr 2673 . . . . . . . 8 ( = 𝑊 → (Base‘) = 𝑉)
87pweqd 4135 . . . . . . 7 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
9 fveq2 6148 . . . . . . . . . . . 12 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
10 isobs.h . . . . . . . . . . . 12 , = (·𝑖𝑊)
119, 10syl6eqr 2673 . . . . . . . . . . 11 ( = 𝑊 → (·𝑖) = , )
1211oveqd 6621 . . . . . . . . . 10 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
13 fveq2 6148 . . . . . . . . . . . . . 14 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
14 isobs.f . . . . . . . . . . . . . 14 𝐹 = (Scalar‘𝑊)
1513, 14syl6eqr 2673 . . . . . . . . . . . . 13 ( = 𝑊 → (Scalar‘) = 𝐹)
1615fveq2d 6152 . . . . . . . . . . . 12 ( = 𝑊 → (1r‘(Scalar‘)) = (1r𝐹))
17 isobs.u . . . . . . . . . . . 12 1 = (1r𝐹)
1816, 17syl6eqr 2673 . . . . . . . . . . 11 ( = 𝑊 → (1r‘(Scalar‘)) = 1 )
1915fveq2d 6152 . . . . . . . . . . . 12 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
20 isobs.z . . . . . . . . . . . 12 0 = (0g𝐹)
2119, 20syl6eqr 2673 . . . . . . . . . . 11 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
2218, 21ifeq12d 4078 . . . . . . . . . 10 ( = 𝑊 → if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) = if(𝑥 = 𝑦, 1 , 0 ))
2312, 22eqeq12d 2636 . . . . . . . . 9 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
24232ralbidv 2983 . . . . . . . 8 ( = 𝑊 → (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
25 fveq2 6148 . . . . . . . . . . 11 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
26 isobs.o . . . . . . . . . . 11 = (ocv‘𝑊)
2725, 26syl6eqr 2673 . . . . . . . . . 10 ( = 𝑊 → (ocv‘) = )
2827fveq1d 6150 . . . . . . . . 9 ( = 𝑊 → ((ocv‘)‘𝑏) = ( 𝑏))
29 fveq2 6148 . . . . . . . . . . 11 ( = 𝑊 → (0g) = (0g𝑊))
30 isobs.y . . . . . . . . . . 11 𝑌 = (0g𝑊)
3129, 30syl6eqr 2673 . . . . . . . . . 10 ( = 𝑊 → (0g) = 𝑌)
3231sneqd 4160 . . . . . . . . 9 ( = 𝑊 → {(0g)} = {𝑌})
3328, 32eqeq12d 2636 . . . . . . . 8 ( = 𝑊 → (((ocv‘)‘𝑏) = {(0g)} ↔ ( 𝑏) = {𝑌}))
3424, 33anbi12d 746 . . . . . . 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 3181 . . . . . 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 6158 . . . . . . . . 9 (Base‘𝑊) ∈ V
376, 36eqeltri 2694 . . . . . . . 8 𝑉 ∈ V
3837pwex 4808 . . . . . . 7 𝒫 𝑉 ∈ V
3938rabex 4773 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})} ∈ V
4035, 1, 39fvmpt 6239 . . . . 5 (𝑊 ∈ PreHil → (OBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})})
4140eleq2d 2684 . . . 4 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})}))
42 raleq 3127 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
4342raleqbi1dv 3135 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
44 fveq2 6148 . . . . . . . 8 (𝑏 = 𝐵 → ( 𝑏) = ( 𝐵))
4544eqeq1d 2623 . . . . . . 7 (𝑏 = 𝐵 → (( 𝑏) = {𝑌} ↔ ( 𝐵) = {𝑌}))
4643, 45anbi12d 746 . . . . . 6 (𝑏 = 𝐵 → ((∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌}) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
4746elrab 3346 . . . . 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 4788 . . . . . 6 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
4948anbi1i 730 . . . . 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 264 . . . 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 276 . . 3 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
524, 51biadan2 673 . 2 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
53 3anass 1040 . 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 267 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555  ifcif 4058  𝒫 cpw 4130  {csn 4148  dom cdm 5074  cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865  ·𝑖cip 15867  0gc0g 16021  1rcur 18422  PreHilcphl 19888  ocvcocv 19923  OBasiscobs 19965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-obs 19968
This theorem is referenced by:  obsip  19984  obsrcl  19986  obsss  19987  obsocv  19989
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