Theorem eldiophb 42849

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 42848	. . . 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 6188	. . 3 ⊢ dom Dioph ⊆ ℕ0
3		elfvdm 6856	. . 3 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
4	2, 3	sselid 3927	. 2 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5		fveq2 6822	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (ℤ≥‘𝑛) = (ℤ≥‘𝑁))
6		eqidd 2732	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7		oveq2 7354	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
8	7	reseq2d 5927	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
9	8	eqeq2d 2742	. . . . . . . . . 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 3156	. . . . . . . 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 2797	. . . . . . 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 7421	. . . . . 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 5877	. . . . 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 7379	. . . . . . 7 ⊢ (ℕ0 ↑m (1...𝑁)) ∈ V
16	15	pwex 5316	. . . . . 6 ⊢ 𝒫 (ℕ0 ↑m (1...𝑁)) ∈ V
17		eqid 2731	. . . . . . . 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 7479	. . . . . . 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 8773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20		fzss2 13464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑘))
21		fssres 6689	. . . . . . . . . . . . . . . . 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 12387	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
24		ovex 7379	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
25	23, 24	elmap 8795	. . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . 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 3133	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
31	30	abssdv 4014	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ (ℕ0 ↑m (1...𝑁)))
32	15	elpw2 5270	. . . . . . . . . . . 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 2819	. . . . . . . . . . 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 3138	. . . . . . . . 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 3126	. . . . . . . 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 4015	. . . . . . 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 3976	. . . . . 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 5258	. . . . 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 6929	. . . 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 2817	. . 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 7379	. . . . . 6 ⊢ (ℕ0 ↑m (1...𝑘)) ∈ V
44	43	abrexex 7894	. . . . 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 3070	. . . . . 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 4013	. . . . 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 5258	. . . 4 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ V
49	17, 48	elrnmpo 7482	. . 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 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4547  dom cdm 5614  ran crn 5615   ↾ cres 5616  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348   ↑_m cmap 8750  0cc0 11006  1c1 11007  ℕ₀cn0 12381  ℤ_≥cuz 12732  ...cfz 13407  mzPolycmzp 42814  Diophcdioph 42847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-addcl 11066  ax-pre-lttri 11080  ax-pre-lttrn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-neg 11347  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-dioph 42848

This theorem is referenced by:  eldioph  42850  eldioph2b  42855  eldiophelnn0  42856