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Theorem eldioph 41367
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 1137 . 2 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑁 ∈ ℕ0)
2 simp2 1138 . . 3 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝐾 ∈ (ℤ𝑁))
3 simp3 1139 . . . 4 ((𝑁 ∈ ℕ0𝐾 ∈ (ℤ𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑃 ∈ (mzPoly‘(1...𝐾)))
4 eqidd 2734 . . . 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 6880 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑢) = (𝑃𝑢))
65eqeq1d 2735 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑢) = 0 ↔ (𝑃𝑢) = 0))
76anbi2d 630 . . . . . . 7 (𝑝 = 𝑃 → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
87rexbidv 3179 . . . . . 6 (𝑝 = 𝑃 → (∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃𝑢) = 0)))
98abbidv 2802 . . . . 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 3631 . . . 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 585 . . 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 7404 . . . . . 6 (𝑘 = 𝐾 → (1...𝑘) = (1...𝐾))
1312fveq2d 6885 . . . . 5 (𝑘 = 𝐾 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝐾)))
1412oveq2d 7412 . . . . . . . 8 (𝑘 = 𝐾 → (ℕ0m (1...𝑘)) = (ℕ0m (1...𝐾)))
1514rexeqdv 3327 . . . . . . 7 (𝑘 = 𝐾 → (∃𝑢 ∈ (ℕ0m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝𝑢) = 0)))
1615abbidv 2802 . . . . . 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 2744 . . . . 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 3344 . . . 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 3611 . . 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 585 . 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 41366 . 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 584 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 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  cres 5674  cfv 6535  (class class class)co 7396  m cmap 8808  0cc0 11097  1c1 11098  0cn0 12459  cuz 12809  ...cfz 13471  mzPolycmzp 41331  Diophcdioph 41364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-addcl 11157  ax-pre-lttri 11171  ax-pre-lttrn 11172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-er 8691  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-neg 11434  df-nn 12200  df-n0 12460  df-z 12546  df-uz 12810  df-fz 13472  df-dioph 41365
This theorem is referenced by:  eldioph2  41371  eq0rabdioph  41385
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