Theorem eldioph 43204

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 6833	. . . . . . . . 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 7368	. . . . . 6 ⊢ (𝑘 = 𝐾 → (1...𝑘) = (1...𝐾))
13	12	fveq2d 6838	. . . . 5 ⊢ (𝑘 = 𝐾 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝐾)))
14	12	oveq2d 7376	. . . . . . . 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 43203	. 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 5626  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  0cc0 11029  1c1 11030  ℕ₀cn0 12428  ℤ_≥cuz 12779  ...cfz 13452  mzPolycmzp 43168  Diophcdioph 43201

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-addcl 11089  ax-pre-lttri 11103  ax-pre-lttrn 11104

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 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-dioph 43202

This theorem is referenced by:  eldioph2  43208  eq0rabdioph  43222