Theorem eldioph 43212

Description: Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)

Assertion

Ref	Expression
eldioph	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 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.

Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑁 ∈ ℕ0)
2		simp2 1138	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝐾 ∈ (ℤ≥‘𝑁))
3		simp3 1139	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑃 ∈ (mzPoly‘(1...𝐾)))
4		eqidd 2738	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)})
5		fveq1 6837	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑢) = (𝑃‘𝑢))
6	5	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑢) = 0 ↔ (𝑃‘𝑢) = 0))
7	6	anbi2d 631	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
8	7	rexbidv 3162	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)))
9	8	abbidv 2803	. . . . 5 ⊢ (𝑝 = 𝑃 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)})
10	9	rspceeqv 3588	. . . 4 ⊢ ((𝑃 ∈ (mzPoly‘(1...𝐾)) ∧ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
11	3, 4, 10	syl2anc 585	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
12		oveq2 7372	. . . . . 6 ⊢ (𝑘 = 𝐾 → (1...𝑘) = (1...𝐾))
13	12	fveq2d 6842	. . . . 5 ⊢ (𝑘 = 𝐾 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝐾)))
14	12	oveq2d 7380	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (ℕ0 ↑m (1...𝑘)) = (ℕ0 ↑m (1...𝐾)))
15	14	rexeqdv 3297	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)))
16	15	abbidv 2803	. . . . . 6 ⊢ (𝑘 = 𝐾 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
17	16	eqeq2d 2748	. . . . 5 ⊢ (𝑘 = 𝐾 → ({𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
18	13, 17	rexeqbidv 3313	. . . 4 ⊢ (𝑘 = 𝐾 → (∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
19	18	rspcev 3565	. . 3 ⊢ ((𝐾 ∈ (ℤ≥‘𝑁) ∧ ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
20	2, 11, 19	syl2anc 585	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
21		eldiophb 43211	. 2 ⊢ ({𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
22	1, 20, 21	sylanbrc 584	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ (ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ↾ cres 5630  ‘cfv 6496  (class class class)co 7364   ↑_m cmap 8770  0cc0 11035  1c1 11036  ℕ₀cn0 12434  ℤ_≥cuz 12785  ...cfz 13458  mzPolycmzp 43176  Diophcdioph 43209

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-addcl 11095  ax-pre-lttri 11109  ax-pre-lttrn 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-neg 11377  df-nn 12172  df-n0 12435  df-z 12522  df-uz 12786  df-fz 13459  df-dioph 43210

This theorem is referenced by:  eldioph2  43216  eq0rabdioph  43230