Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldiophb Structured version   Visualization version   GIF version

Theorem eldiophb 37840
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 37839 . . . 4 Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0)}))
21dmmptss 5859 . . 3 dom Dioph ⊆ ℕ0
3 elfvdm 6450 . . 3 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
42, 3sseldi 3807 . 2 (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5 fveq2 6418 . . . . . . 7 (𝑛 = 𝑁 → (ℤ𝑛) = (ℤ𝑁))
6 eqidd 2818 . . . . . . 7 (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7 oveq2 6892 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
87reseq2d 5611 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
98eqeq2d 2827 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
109anbi1d 617 . . . . . . . . 9 (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1110rexbidv 3251 . . . . . . . 8 (𝑛 = 𝑁 → (∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1211abbidv 2936 . . . . . . 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, 12mpt2eq123dv 6957 . . . . . 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 5568 . . . . 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 6916 . . . . . . 7 (ℕ0𝑚 (1...𝑁)) ∈ V
1615pwex 5063 . . . . . 6 𝒫 (ℕ0𝑚 (1...𝑁)) ∈ V
17 eqid 2817 . . . . . . . 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)})
1817rnmpt2 7010 . . . . . . 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 8124 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (ℕ0𝑚 (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20 fzss2 12624 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑘))
21 fssres 6295 . . . . . . . . . . . . . . . . 17 ((𝑢:(1...𝑘)⟶ℕ0 ∧ (1...𝑁) ⊆ (1...𝑘)) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2219, 20, 21syl2anr 586 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0𝑚 (1...𝑘))) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
23 nn0ex 11585 . . . . . . . . . . . . . . . . 17 0 ∈ V
24 ovex 6916 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
2523, 24elmap 8131 . . . . . . . . . . . . . . . 16 ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
2622, 25sylibr 225 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0𝑚 (1...𝑘))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
27 eleq1 2884 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁))))
2827adantr 468 . . . . . . . . . . . . . . 15 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → (𝑡 ∈ (ℕ0𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁))))
2926, 28syl5ibrcom 238 . . . . . . . . . . . . . 14 ((𝑘 ∈ (ℤ𝑁) ∧ 𝑢 ∈ (ℕ0𝑚 (1...𝑘))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 ∈ (ℕ0𝑚 (1...𝑁))))
3029rexlimdva 3230 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ𝑁) → (∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 ∈ (ℕ0𝑚 (1...𝑁))))
3130abssdv 3884 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ (ℕ0𝑚 (1...𝑁)))
3215elpw2 5033 . . . . . . . . . . . 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 225 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ 𝒫 (ℕ0𝑚 (1...𝑁)))
34 eleq1 2884 . . . . . . . . . . 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 238 . . . . . . . . . 10 (𝑘 ∈ (ℤ𝑁) → (𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0𝑚 (1...𝑁))))
3635rexlimdvw 3233 . . . . . . . . 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 3226 . . . . . . . 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 3885 . . . . . . 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 3843 . . . . . 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 5011 . . . . 5 ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}) ∈ V
4114, 1, 40fvmpt 6513 . . . 4 (𝑁 ∈ ℕ0 → (Dioph‘𝑁) = ran (𝑘 ∈ (ℤ𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
4241eleq2d 2882 . . 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 6916 . . . . . 6 (ℕ0𝑚 (1...𝑘)) ∈ V
4443abrexex 7381 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))} ∈ V
45 simpl 470 . . . . . . 7 ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → 𝑡 = (𝑢 ↾ (1...𝑁)))
4645reximi 3209 . . . . . 6 (∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) → ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁)))
4746ss2abi 3882 . . . . 5 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ⊆ {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))}
4844, 47ssexi 5011 . . . 4 {𝑡 ∣ ∃𝑢 ∈ (ℕ0𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ∈ V
4917, 48elrnmpt2 7013 . . 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, 49syl6bb 278 . 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, 50biadan2 844 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 197  wa 384   = wceq 1637  wcel 2157  {cab 2803  wrex 3108  wss 3780  𝒫 cpw 4362  dom cdm 5324  ran crn 5325  cres 5326  wf 6107  cfv 6111  (class class class)co 6884  cmpt2 6886  𝑚 cmap 8102  0cc0 10231  1c1 10232  0cn0 11579  cuz 11924  ...cfz 12569  mzPolycmzp 37805  Diophcdioph 37838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-addcl 10291  ax-pre-lttri 10305  ax-pre-lttrn 10306
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-er 7989  df-map 8104  df-en 8203  df-dom 8204  df-sdom 8205  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-neg 10564  df-nn 11316  df-n0 11580  df-z 11664  df-uz 11925  df-fz 12570  df-dioph 37839
This theorem is referenced by:  eldioph  37841  eldioph2b  37846  eldiophelnn0  37847
  Copyright terms: Public domain W3C validator