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Theorem eldioph 39233
Description: Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
Assertion
Ref Expression
eldioph ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑡,𝑁,𝑢   𝑡,𝐾,𝑢   𝑡,𝑃,𝑢

Proof of Theorem eldioph
Dummy variables 𝑘 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1128 . 2 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑁 ∈ ℕ0)
2 simp2 1129 . . 3 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝐾 ∈ (ℤ𝑁))
3 simp3 1130 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑃 ∈ (mzPoly‘(1...𝐾)))
4 eqidd 2819 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)})
5 fveq1 6662 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑢) = (𝑃𝑢))
65eqeq1d 2820 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑢) = 0 ↔ (𝑃𝑢) = 0))
76anbi2d 628 . . . . . . 7 (𝑝 = 𝑃 → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
87rexbidv 3294 . . . . . 6 (𝑝 = 𝑃 → (∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
98abbidv 2882 . . . . 5 (𝑝 = 𝑃 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)})
109rspceeqv 3635 . . . 4 ((𝑃 ∈ (mzPoly‘(1...𝐾)) ∧ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
113, 4, 10syl2anc 584 . . 3 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
12 oveq2 7153 . . . . . 6 (𝑘 = 𝐾 → (1...𝑘) = (1...𝐾))
1312fveq2d 6667 . . . . 5 (𝑘 = 𝐾 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝐾)))
1412oveq2d 7161 . . . . . . . 8 (𝑘 = 𝐾 → (ℕ0m (1...𝑘)) = (ℕ0m (1...𝐾)))
1514rexeqdv 3414 . . . . . . 7 (𝑘 = 𝐾 → (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1615abbidv 2882 . . . . . 6 (𝑘 = 𝐾 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
1716eqeq2d 2829 . . . . 5 (𝑘 = 𝐾 → ({𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
1813, 17rexeqbidv 3400 . . . 4 (𝑘 = 𝐾 → (∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
1918rspcev 3620 . . 3 ((𝐾 ∈ (ℤ𝑁) ∧ ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {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)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
202, 11, 19syl2anc 584 . 2 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)})
21 eldiophb 39232 . 2 ({𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)}))
221, 20, 21sylanbrc 583 1 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  cres 5550  cfv 6348  (class class class)co 7145  m cmap 8395  0cc0 10525  1c1 10526  0cn0 11885  cuz 12231  ...cfz 12880  mzPolycmzp 39197  Diophcdioph 39230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-addcl 10585  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-dioph 39231
This theorem is referenced by:  eldioph2  39237  eq0rabdioph  39251
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