Theorem eldiophb 42750

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 42749	. . . 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 6190	. . 3 ⊢ dom Dioph ⊆ ℕ0
3		elfvdm 6857	. . 3 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
4	2, 3	sselid 3933	. 2 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5		fveq2 6822	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (ℤ≥‘𝑛) = (ℤ≥‘𝑁))
6		eqidd 2730	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7		oveq2 7357	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
8	7	reseq2d 5930	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
9	8	eqeq2d 2740	. . . . . . . . . 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 3153	. . . . . . . 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 2795	. . . . . . 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 7424	. . . . . 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 5880	. . . . 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 7382	. . . . . . 7 ⊢ (ℕ0 ↑m (1...𝑁)) ∈ V
16	15	pwex 5319	. . . . . 6 ⊢ 𝒫 (ℕ0 ↑m (1...𝑁)) ∈ V
17		eqid 2729	. . . . . . . 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 7482	. . . . . . 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 8776	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20		fzss2 13467	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑘))
21		fssres 6690	. . . . . . . . . . . . . . . . 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 12390	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
24		ovex 7382	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
25	23, 24	elmap 8798	. . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
31	30	abssdv 4020	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ (ℕ0 ↑m (1...𝑁)))
32	15	elpw2 5273	. . . . . . . . . . . 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 2816	. . . . . . . . . . 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 3135	. . . . . . . . 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 3123	. . . . . . . 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 4021	. . . . . . 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 3982	. . . . . 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 5261	. . . . 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 6930	. . . 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 2814	. . 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 7382	. . . . . 6 ⊢ (ℕ0 ↑m (1...𝑘)) ∈ V
44	43	abrexex 7897	. . . . 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 3067	. . . . . 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 4019	. . . . 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 5261	. . . 4 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ V
49	17, 48	elrnmpo 7485	. . 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 2707  ∃wrex 3053   ⊆ wss 3903  𝒫 cpw 4551  dom cdm 5619  ran crn 5620   ↾ cres 5621  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753  0cc0 11009  1c1 11010  ℕ₀cn0 12384  ℤ_≥cuz 12735  ...cfz 13410  mzPolycmzp 42715  Diophcdioph 42748

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-addcl 11069  ax-pre-lttri 11083  ax-pre-lttrn 11084

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-dioph 42749

This theorem is referenced by:  eldioph  42751  eldioph2b  42756  eldiophelnn0  42757