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Theorem isobs 20282
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 20267 . . . . 5 OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
21dmmptss 5776 . . . 4 dom OBasis ⊆ PreHil
3 elfvdm 6362 . . . 4 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ dom OBasis)
42, 3sseldi 3751 . . 3 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
5 fveq2 6333 . . . . . . . . 9 ( = 𝑊 → (Base‘) = (Base‘𝑊))
6 isobs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
75, 6syl6eqr 2823 . . . . . . . 8 ( = 𝑊 → (Base‘) = 𝑉)
87pweqd 4303 . . . . . . 7 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
9 fveq2 6333 . . . . . . . . . . . 12 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
10 isobs.h . . . . . . . . . . . 12 , = (·𝑖𝑊)
119, 10syl6eqr 2823 . . . . . . . . . . 11 ( = 𝑊 → (·𝑖) = , )
1211oveqd 6811 . . . . . . . . . 10 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
13 fveq2 6333 . . . . . . . . . . . . . 14 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
14 isobs.f . . . . . . . . . . . . . 14 𝐹 = (Scalar‘𝑊)
1513, 14syl6eqr 2823 . . . . . . . . . . . . 13 ( = 𝑊 → (Scalar‘) = 𝐹)
1615fveq2d 6337 . . . . . . . . . . . 12 ( = 𝑊 → (1r‘(Scalar‘)) = (1r𝐹))
17 isobs.u . . . . . . . . . . . 12 1 = (1r𝐹)
1816, 17syl6eqr 2823 . . . . . . . . . . 11 ( = 𝑊 → (1r‘(Scalar‘)) = 1 )
1915fveq2d 6337 . . . . . . . . . . . 12 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
20 isobs.z . . . . . . . . . . . 12 0 = (0g𝐹)
2119, 20syl6eqr 2823 . . . . . . . . . . 11 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
2218, 21ifeq12d 4246 . . . . . . . . . 10 ( = 𝑊 → if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) = if(𝑥 = 𝑦, 1 , 0 ))
2312, 22eqeq12d 2786 . . . . . . . . 9 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
24232ralbidv 3138 . . . . . . . 8 ( = 𝑊 → (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
25 fveq2 6333 . . . . . . . . . . 11 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
26 isobs.o . . . . . . . . . . 11 = (ocv‘𝑊)
2725, 26syl6eqr 2823 . . . . . . . . . 10 ( = 𝑊 → (ocv‘) = )
2827fveq1d 6335 . . . . . . . . 9 ( = 𝑊 → ((ocv‘)‘𝑏) = ( 𝑏))
29 fveq2 6333 . . . . . . . . . . 11 ( = 𝑊 → (0g) = (0g𝑊))
30 isobs.y . . . . . . . . . . 11 𝑌 = (0g𝑊)
3129, 30syl6eqr 2823 . . . . . . . . . 10 ( = 𝑊 → (0g) = 𝑌)
3231sneqd 4329 . . . . . . . . 9 ( = 𝑊 → {(0g)} = {𝑌})
3328, 32eqeq12d 2786 . . . . . . . 8 ( = 𝑊 → (((ocv‘)‘𝑏) = {(0g)} ↔ ( 𝑏) = {𝑌}))
3424, 33anbi12d 610 . . . . . . 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 3345 . . . . . 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 6343 . . . . . . . . 9 (Base‘𝑊) ∈ V
376, 36eqeltri 2846 . . . . . . . 8 𝑉 ∈ V
3837pwex 4982 . . . . . . 7 𝒫 𝑉 ∈ V
3938rabex 4947 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})} ∈ V
4035, 1, 39fvmpt 6425 . . . . 5 (𝑊 ∈ PreHil → (OBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})})
4140eleq2d 2836 . . . 4 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})}))
42 raleq 3287 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
4342raleqbi1dv 3295 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
44 fveq2 6333 . . . . . . . 8 (𝑏 = 𝐵 → ( 𝑏) = ( 𝐵))
4544eqeq1d 2773 . . . . . . 7 (𝑏 = 𝐵 → (( 𝑏) = {𝑌} ↔ ( 𝐵) = {𝑌}))
4643, 45anbi12d 610 . . . . . 6 (𝑏 = 𝐵 → ((∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌}) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
4746elrab 3516 . . . . 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 4960 . . . . . 6 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
4948anbi1i 604 . . . . 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 813 . 2 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
53 3anass 1080 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  wss 3724  ifcif 4226  𝒫 cpw 4298  {csn 4317  dom cdm 5250  cfv 6032  (class class class)co 6794  Basecbs 16065  Scalarcsca 16153  ·𝑖cip 16155  0gc0g 16309  1rcur 18710  PreHilcphl 20187  ocvcocv 20222  OBasiscobs 20264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fv 6040  df-ov 6797  df-obs 20267
This theorem is referenced by:  obsip  20283  obsrcl  20285  obsss  20286  obsocv  20288
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