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Theorem eldiophb 42744
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 42743 . . . 4 Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}))
21dmmptss 6262 . . 3 dom Dioph ⊆ ℕ0
3 elfvdm 6943 . . 3 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
42, 3sselid 3992 . 2 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5 fveq2 6906 . . . . . . 7 (𝑛 = 𝑁 → (ℤ𝑛) = (ℤ𝑁))
6 eqidd 2735 . . . . . . 7 (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7 oveq2 7438 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
87reseq2d 5999 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
98eqeq2d 2745 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
109anbi1d 631 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1110rexbidv 3176 . . . . . . . 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 7507 . . . . . 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 5951 . . . . 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 7463 . . . . . . 7 (ℕ0m (1...𝑁)) ∈ V
1615pwex 5385 . . . . . 6 𝒫 (ℕ0m (1...𝑁)) ∈ V
17 eqid 2734 . . . . . . . 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 7565 . . . . . . 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 8887 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20 fzss2 13600 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑘))
21 fssres 6774 . . . . . . . . . . . . . . . . 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 12529 . . . . . . . . . . . . . . . . 17 0 ∈ V
24 ovex 7463 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
2523, 24elmap 8909 . . . . . . . . . . . . . . . 16 ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2622, 25sylibr 234 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁))))
2926, 28syl5ibrcom 247 . . . . . . . . . . . . . 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 4077 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ (ℕ0m (1...𝑁)))
3215elpw2 5339 . . . . . . . . . . . 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 234 . . . . . . . . . . 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 247 . . . . . . . . . 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 4079 . . . . . . 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 4029 . . . . . 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 5327 . . . . 5 ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) ∈ V
4114, 1, 40fvmpt 7015 . . . 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 7463 . . . . . 6 (ℕ0m (1...𝑘)) ∈ V
4443abrexex 7985 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))} ∈ V
45 simpl 482 . . . . . . 7 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 = (𝑢 ↾ (1...𝑁)))
4645reximi 3081 . . . . . 6 (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁)))
4746ss2abi 4076 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))}
4844, 47ssexi 5327 . . . 4 {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ V
4917, 48elrnmpo 7568 . . 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 287 . 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 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  wrex 3067  wss 3962  𝒫 cpw 4604  dom cdm 5688  ran crn 5689  cres 5690  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  m cmap 8864  0cc0 11152  1c1 11153  0cn0 12523  cuz 12875  ...cfz 13543  mzPolycmzp 42709  Diophcdioph 42742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-addcl 11212  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-dioph 42743
This theorem is referenced by:  eldioph  42745  eldioph2b  42750  eldiophelnn0  42751
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