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Theorem eldiophb 41066
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 41065 . . . 4 Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}))
21dmmptss 6193 . . 3 dom Dioph ⊆ ℕ0
3 elfvdm 6879 . . 3 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
42, 3sselid 3942 . 2 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5 fveq2 6842 . . . . . . 7 (𝑛 = 𝑁 → (ℤ𝑛) = (ℤ𝑁))
6 eqidd 2737 . . . . . . 7 (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7 oveq2 7365 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
87reseq2d 5937 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
98eqeq2d 2747 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
109anbi1d 630 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1110rexbidv 3175 . . . . . . . 8 (𝑛 = 𝑁 → (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1211abbidv 2805 . . . . . . 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 7432 . . . . . 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 5893 . . . . 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 7390 . . . . . . 7 (ℕ0m (1...𝑁)) ∈ V
1615pwex 5335 . . . . . 6 𝒫 (ℕ0m (1...𝑁)) ∈ V
17 eqid 2736 . . . . . . . 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 7489 . . . . . . 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 8787 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20 fzss2 13481 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑘))
21 fssres 6708 . . . . . . . . . . . . . . . . 17 ((𝑢:(1...𝑘)⟶ℕ0 ∧ (1...𝑁) ⊆ (1...𝑘)) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2219, 20, 21syl2anr 597 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0m (1...𝑘))) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
23 nn0ex 12419 . . . . . . . . . . . . . . . . 17 0 ∈ V
24 ovex 7390 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
2523, 24elmap 8809 . . . . . . . . . . . . . . . 16 ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2622, 25sylibr 233 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0m (1...𝑘))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)))
27 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁))))
2827adantr 481 . . . . . . . . . . . . . . 15 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁))))
2926, 28syl5ibrcom 246 . . . . . . . . . . . . . 14 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0m (1...𝑘))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 ∈ (ℕ0m (1...𝑁))))
3029rexlimdva 3152 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ𝑁) → (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 ∈ (ℕ0m (1...𝑁))))
3130abssdv 4025 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ (ℕ0m (1...𝑁)))
3215elpw2 5302 . . . . . . . . . . . 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 233 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ 𝒫 (ℕ0m (1...𝑁)))
34 eleq1 2825 . . . . . . . . . . 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 246 . . . . . . . . . 10 (𝑘 ∈ (ℤ𝑁) → (𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0m (1...𝑁))))
3635rexlimdvw 3157 . . . . . . . . 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 3145 . . . . . . . 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 4027 . . . . . . 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 3978 . . . . . 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 5279 . . . . 5 ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) ∈ V
4114, 1, 40fvmpt 6948 . . . 4 (𝑁 ∈ ℕ0 → (Dioph‘𝑁) = ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
4241eleq2d 2823 . . 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 7390 . . . . . 6 (ℕ0m (1...𝑘)) ∈ V
4443abrexex 7895 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))} ∈ V
45 simpl 483 . . . . . . 7 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 = (𝑢 ↾ (1...𝑁)))
4645reximi 3087 . . . . . 6 (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁)))
4746ss2abi 4023 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))}
4844, 47ssexi 5279 . . . 4 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ V
4917, 48elrnmpo 7492 . . 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 286 . 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 820 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 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  wss 3910  𝒫 cpw 4560  dom cdm 5633  ran crn 5634  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  0cc0 11051  1c1 11052  0cn0 12413  cuz 12763  ...cfz 13424  mzPolycmzp 41031  Diophcdioph 41064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-addcl 11111  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-dioph 41065
This theorem is referenced by:  eldioph  41067  eldioph2b  41072  eldiophelnn0  41073
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