Theorem eldiophb 42752

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 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 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.

Step	Hyp	Ref	Expression
1		df-dioph 42751	. . . 4 ⊢ Dioph = (𝑛 ∈ ℕ0 ↦ ran (𝑘 ∈ (ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)}))
2	1	dmmptss 6217	. . 3 ⊢ dom Dioph ⊆ ℕ0
3		elfvdm 6898	. . 3 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
4	2, 3	sselid 3947	. 2 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5		fveq2 6861	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (ℤ≥‘𝑛) = (ℤ≥‘𝑁))
6		eqidd 2731	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7		oveq2 7398	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
8	7	reseq2d 5953	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
9	8	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
10	9	anbi1d 631	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)))
11	10	rexbidv 3158	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)))
12	11	abbidv 2796	. . . . . . 7 ⊢ (𝑛 = 𝑁 → {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)})
13	5, 6, 12	mpoeq123dv 7467	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑘 ∈ (ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {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)}))
14	13	rneqd 5905	. . . . 5 ⊢ (𝑛 = 𝑁 → ran (𝑘 ∈ (ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)}) = ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
15		ovex 7423	. . . . . . 7 ⊢ (ℕ0 ↑m (1...𝑁)) ∈ V
16	15	pwex 5338	. . . . . 6 ⊢ 𝒫 (ℕ0 ↑m (1...𝑁)) ∈ V
17		eqid 2730	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {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)})
18	17	rnmpo 7525	. . . . . . 7 ⊢ ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) = {𝑑 ∣ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}}
19		elmapi 8825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20		fzss2 13532	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑘))
21		fssres 6729	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢:(1...𝑘)⟶ℕ0 ∧ (1...𝑁) ⊆ (1...𝑘)) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
22	19, 20, 21	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (ℤ≥‘𝑁) ∧ 𝑢 ∈ (ℕ0 ↑m (1...𝑘))) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
23		nn0ex 12455	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
24		ovex 7423	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
25	23, 24	elmap 8847	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
26	22, 25	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (ℤ≥‘𝑁) ∧ 𝑢 ∈ (ℕ0 ↑m (1...𝑘))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
27		eleq1 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
28	27	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
29	26, 28	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ≥‘𝑁) ∧ 𝑢 ∈ (ℕ0 ↑m (1...𝑘))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
30	29	rexlimdva 3135	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
31	30	abssdv 4034	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ (ℕ0 ↑m (1...𝑁)))
32	15	elpw2 5292	. . . . . . . . . . . 12 ⊢ ({𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ 𝒫 (ℕ0 ↑m (1...𝑁)) ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ (ℕ0 ↑m (1...𝑁)))
33	31, 32	sylibr 234	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ 𝒫 (ℕ0 ↑m (1...𝑁)))
34		eleq1 2817	. . . . . . . . . . 11 ⊢ (𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} → (𝑑 ∈ 𝒫 (ℕ0 ↑m (1...𝑁)) ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ 𝒫 (ℕ0 ↑m (1...𝑁))))
35	33, 34	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0 ↑m (1...𝑁))))
36	35	rexlimdvw 3140	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0 ↑m (1...𝑁))))
37	36	rexlimiv 3128	. . . . . . . 8 ⊢ (∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} → 𝑑 ∈ 𝒫 (ℕ0 ↑m (1...𝑁)))
38	37	abssi 4036	. . . . . . 7 ⊢ {𝑑 ∣ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝑑 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}} ⊆ 𝒫 (ℕ0 ↑m (1...𝑁))
39	18, 38	eqsstri 3996	. . . . . 6 ⊢ ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) ⊆ 𝒫 (ℕ0 ↑m (1...𝑁))
40	16, 39	ssexi 5280	. . . . 5 ⊢ ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) ∈ V
41	14, 1, 40	fvmpt 6971	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (Dioph‘𝑁) = ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
42	41	eleq2d 2815	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐷 ∈ (Dioph‘𝑁) ↔ 𝐷 ∈ ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})))
43		ovex 7423	. . . . . 6 ⊢ (ℕ0 ↑m (1...𝑘)) ∈ V
44	43	abrexex 7944	. . . . 5 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))} ∈ V
45		simpl 482	. . . . . . 7 ⊢ ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 = (𝑢 ↾ (1...𝑁)))
46	45	reximi 3068	. . . . . 6 ⊢ (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁)))
47	46	ss2abi 4033	. . . . 5 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))}
48	44, 47	ssexi 5280	. . . 4 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ V
49	17, 48	elrnmpo 7528	. . 3 ⊢ (𝐷 ∈ ran (𝑘 ∈ (ℤ≥‘𝑁), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) ↔ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})
50	42, 49	bitrdi 287	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝐷 ∈ (Dioph‘𝑁) ↔ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
51	4, 50	biadanii 821	1 ⊢ (𝐷 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘))𝐷 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054   ⊆ wss 3917  𝒫 cpw 4566  dom cdm 5641  ran crn 5642   ↾ cres 5643  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ↑_m cmap 8802  0cc0 11075  1c1 11076  ℕ₀cn0 12449  ℤ_≥cuz 12800  ...cfz 13475  mzPolycmzp 42717  Diophcdioph 42750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-addcl 11135  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-neg 11415  df-nn 12194  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-dioph 42751

This theorem is referenced by:  eldioph  42753  eldioph2b  42758  eldiophelnn0  42759