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Theorem eldiophb 40310
Description: Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Assertion
Ref Expression
eldiophb (𝐷 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
Distinct variable groups:   𝐷,𝑘,𝑝   𝑘,𝑁,𝑝,𝑡,𝑢
Allowed substitution hints:   𝐷(𝑢,𝑡)

Proof of Theorem eldiophb
Dummy variables 𝑛 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dioph 40309 . . . 4 Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}))
21dmmptss 6118 . . 3 dom Dioph ⊆ ℕ0
3 elfvdm 6767 . . 3 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
42, 3sseldi 3913 . 2 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5 fveq2 6735 . . . . . . 7 (𝑛 = 𝑁 → (ℤ𝑛) = (ℤ𝑁))
6 eqidd 2739 . . . . . . 7 (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7 oveq2 7239 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
87reseq2d 5865 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
98eqeq2d 2749 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
109anbi1d 633 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1110rexbidv 3224 . . . . . . . 8 (𝑛 = 𝑁 → (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1211abbidv 2808 . . . . . . 7 (𝑛 = 𝑁 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
135, 6, 12mpoeq123dv 7304 . . . . . 6 (𝑛 = 𝑁 → (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}) = (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
1413rneqd 5821 . . . . 5 (𝑛 = 𝑁 → ran (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}) = ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
15 ovex 7264 . . . . . . 7 (ℕ0m (1...𝑁)) ∈ V
1615pwex 5287 . . . . . 6 𝒫 (ℕ0m (1...𝑁)) ∈ V
17 eqid 2738 . . . . . . . 8 (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) = (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
1817rnmpo 7361 . . . . . . 7 ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) = {𝑑 ∣ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}}
19 elmapi 8550 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20 fzss2 13176 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑘))
21 fssres 6603 . . . . . . . . . . . . . . . . 17 ((𝑢:(1...𝑘)⟶ℕ0 ∧ (1...𝑁) ⊆ (1...𝑘)) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2219, 20, 21syl2anr 600 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0m (1...𝑘))) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
23 nn0ex 12120 . . . . . . . . . . . . . . . . 17 0 ∈ V
24 ovex 7264 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
2523, 24elmap 8572 . . . . . . . . . . . . . . . 16 ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2622, 25sylibr 237 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0m (1...𝑘))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)))
27 eleq1 2826 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁))))
2827adantr 484 . . . . . . . . . . . . . . 15 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁))))
2926, 28syl5ibrcom 250 . . . . . . . . . . . . . 14 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0m (1...𝑘))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 ∈ (ℕ0m (1...𝑁))))
3029rexlimdva 3211 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ𝑁) → (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 ∈ (ℕ0m (1...𝑁))))
3130abssdv 3996 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ (ℕ0m (1...𝑁)))
3215elpw2 5252 . . . . . . . . . . . 12 ({𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ 𝒫 (ℕ0m (1...𝑁)) ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ (ℕ0m (1...𝑁)))
3331, 32sylibr 237 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ 𝒫 (ℕ0m (1...𝑁)))
34 eleq1 2826 . . . . . . . . . . 11 (𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → (𝑑 ∈ 𝒫 (ℕ0m (1...𝑁)) ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ 𝒫 (ℕ0m (1...𝑁))))
3533, 34syl5ibrcom 250 . . . . . . . . . 10 (𝑘 ∈ (ℤ𝑁) → (𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0m (1...𝑁))))
3635rexlimdvw 3217 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑁) → (∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0m (1...𝑁))))
3736rexlimiv 3207 . . . . . . . 8 (∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0m (1...𝑁)))
3837abssi 3997 . . . . . . 7 {𝑑 ∣ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}} ⊆ 𝒫 (ℕ0m (1...𝑁))
3918, 38eqsstri 3949 . . . . . 6 ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) ⊆ 𝒫 (ℕ0m (1...𝑁))
4016, 39ssexi 5229 . . . . 5 ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) ∈ V
4114, 1, 40fvmpt 6836 . . . 4 (𝑁 ∈ ℕ0 → (Dioph‘𝑁) = ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
4241eleq2d 2824 . . 3 (𝑁 ∈ ℕ0 → (𝐷 ∈ (Dioph‘𝑁) ↔ 𝐷 ∈ ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})))
43 ovex 7264 . . . . . 6 (ℕ0m (1...𝑘)) ∈ V
4443abrexex 7753 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))} ∈ V
45 simpl 486 . . . . . . 7 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 = (𝑢 ↾ (1...𝑁)))
4645reximi 3173 . . . . . 6 (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁)))
4746ss2abi 3994 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))}
4844, 47ssexi 5229 . . . 4 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ V
4917, 48elrnmpo 7364 . . 3 (𝐷 ∈ ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) ↔ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
5042, 49bitrdi 290 . 2 (𝑁 ∈ ℕ0 → (𝐷 ∈ (Dioph‘𝑁) ↔ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
514, 50biadanii 822 1 (𝐷 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2111  {cab 2715  wrex 3063  wss 3880  𝒫 cpw 4527  dom cdm 5565  ran crn 5566  cres 5567  wf 6393  cfv 6397  (class class class)co 7231  cmpo 7233  m cmap 8528  0cc0 10753  1c1 10754  0cn0 12114  cuz 12462  ...cfz 13119  mzPolycmzp 40275  Diophcdioph 40308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5272  ax-pr 5336  ax-un 7541  ax-cnex 10809  ax-resscn 10810  ax-1cn 10811  ax-addcl 10813  ax-pre-lttri 10827  ax-pre-lttrn 10828
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3422  df-sbc 3709  df-csb 3826  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-pss 3899  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4834  df-iun 4920  df-br 5068  df-opab 5130  df-mpt 5150  df-tr 5176  df-id 5469  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-pred 6175  df-ord 6233  df-on 6234  df-lim 6235  df-suc 6236  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-f1 6402  df-fo 6403  df-f1o 6404  df-fv 6405  df-ov 7234  df-oprab 7235  df-mpo 7236  df-om 7663  df-1st 7779  df-2nd 7780  df-wrecs 8067  df-recs 8128  df-rdg 8166  df-er 8411  df-map 8530  df-en 8647  df-dom 8648  df-sdom 8649  df-pnf 10893  df-mnf 10894  df-xr 10895  df-ltxr 10896  df-le 10897  df-neg 11089  df-nn 11855  df-n0 12115  df-z 12201  df-uz 12463  df-fz 13120  df-dioph 40309
This theorem is referenced by:  eldioph  40311  eldioph2b  40316  eldiophelnn0  40317
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