MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isobs Structured version   Visualization version   GIF version

Theorem isobs 21248
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 21233 . . . 4 OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
21mptrcl 6996 . . 3 (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
3 fveq2 6881 . . . . . . . . 9 ( = 𝑊 → (Base‘) = (Base‘𝑊))
4 isobs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
53, 4eqtr4di 2791 . . . . . . . 8 ( = 𝑊 → (Base‘) = 𝑉)
65pweqd 4615 . . . . . . 7 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
7 fveq2 6881 . . . . . . . . . . . 12 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
8 isobs.h . . . . . . . . . . . 12 , = (·𝑖𝑊)
97, 8eqtr4di 2791 . . . . . . . . . . 11 ( = 𝑊 → (·𝑖) = , )
109oveqd 7413 . . . . . . . . . 10 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
11 fveq2 6881 . . . . . . . . . . . . . 14 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
12 isobs.f . . . . . . . . . . . . . 14 𝐹 = (Scalar‘𝑊)
1311, 12eqtr4di 2791 . . . . . . . . . . . . 13 ( = 𝑊 → (Scalar‘) = 𝐹)
1413fveq2d 6885 . . . . . . . . . . . 12 ( = 𝑊 → (1r‘(Scalar‘)) = (1r𝐹))
15 isobs.u . . . . . . . . . . . 12 1 = (1r𝐹)
1614, 15eqtr4di 2791 . . . . . . . . . . 11 ( = 𝑊 → (1r‘(Scalar‘)) = 1 )
1713fveq2d 6885 . . . . . . . . . . . 12 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
18 isobs.z . . . . . . . . . . . 12 0 = (0g𝐹)
1917, 18eqtr4di 2791 . . . . . . . . . . 11 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
2016, 19ifeq12d 4545 . . . . . . . . . 10 ( = 𝑊 → if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) = if(𝑥 = 𝑦, 1 , 0 ))
2110, 20eqeq12d 2749 . . . . . . . . 9 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
22212ralbidv 3219 . . . . . . . 8 ( = 𝑊 → (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
23 fveq2 6881 . . . . . . . . . . 11 ( = 𝑊 → (ocv‘) = (ocv‘𝑊))
24 isobs.o . . . . . . . . . . 11 = (ocv‘𝑊)
2523, 24eqtr4di 2791 . . . . . . . . . 10 ( = 𝑊 → (ocv‘) = )
2625fveq1d 6883 . . . . . . . . 9 ( = 𝑊 → ((ocv‘)‘𝑏) = ( 𝑏))
27 fveq2 6881 . . . . . . . . . . 11 ( = 𝑊 → (0g) = (0g𝑊))
28 isobs.y . . . . . . . . . . 11 𝑌 = (0g𝑊)
2927, 28eqtr4di 2791 . . . . . . . . . 10 ( = 𝑊 → (0g) = 𝑌)
3029sneqd 4636 . . . . . . . . 9 ( = 𝑊 → {(0g)} = {𝑌})
3126, 30eqeq12d 2749 . . . . . . . 8 ( = 𝑊 → (((ocv‘)‘𝑏) = {(0g)} ↔ ( 𝑏) = {𝑌}))
3222, 31anbi12d 632 . . . . . . 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})))
336, 32rabeqbidv 3450 . . . . . 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})})
344fvexi 6895 . . . . . . . 8 𝑉 ∈ V
3534pwex 5374 . . . . . . 7 𝒫 𝑉 ∈ V
3635rabex 5328 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})} ∈ V
3733, 1, 36fvmpt 6987 . . . . 5 (𝑊 ∈ PreHil → (OBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})})
3837eleq2d 2820 . . . 4 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌})}))
39 raleq 3323 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
4039raleqbi1dv 3334 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 )))
41 fveqeq2 6890 . . . . . . 7 (𝑏 = 𝐵 → (( 𝑏) = {𝑌} ↔ ( 𝐵) = {𝑌}))
4240, 41anbi12d 632 . . . . . 6 (𝑏 = 𝐵 → ((∀𝑥𝑏𝑦𝑏 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝑏) = {𝑌}) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
4342elrab 3681 . . . . 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})))
4434elpw2 5341 . . . . . 6 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
4544anbi1i 625 . . . . 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})))
4643, 45bitri 275 . . . 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})))
4738, 46bitrdi 287 . . 3 (𝑊 ∈ PreHil → (𝐵 ∈ (OBasis‘𝑊) ↔ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
482, 47biadanii 821 . 2 (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ (𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌}))))
49 3anass 1096 . 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}))))
5048, 49bitr4i 278 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  {crab 3433  wss 3946  ifcif 4524  𝒫 cpw 4598  {csn 4624  cfv 6535  (class class class)co 7396  Basecbs 17131  Scalarcsca 17187  ·𝑖cip 17189  0gc0g 17372  1rcur 19987  PreHilcphl 21150  ocvcocv 21186  OBasiscobs 21230
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 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-obs 21233
This theorem is referenced by:  obsip  21249  obsrcl  21251  obsss  21252  obsocv  21254
  Copyright terms: Public domain W3C validator