Theorem eldiophb 40579

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 40578	. . . 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 6144	. . 3 ⊢ dom Dioph ⊆ ℕ0
3		elfvdm 6806	. . 3 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
4	2, 3	sselid 3919	. 2 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5		fveq2 6774	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (ℤ≥‘𝑛) = (ℤ≥‘𝑁))
6		eqidd 2739	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7		oveq2 7283	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
8	7	reseq2d 5891	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
9	8	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
10	9	anbi1d 630	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)))
11	10	rexbidv 3226	. . . . . . . 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 2807	. . . . . . 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 7350	. . . . . 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 5847	. . . . 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 7308	. . . . . . 7 ⊢ (ℕ0 ↑m (1...𝑁)) ∈ V
16	15	pwex 5303	. . . . . 6 ⊢ 𝒫 (ℕ0 ↑m (1...𝑁)) ∈ V
17		eqid 2738	. . . . . . . 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 7407	. . . . . . 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 8637	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20		fzss2 13296	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑘))
21		fssres 6640	. . . . . . . . . . . . . . . . 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 12239	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
24		ovex 7308	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
25	23, 24	elmap 8659	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
26	22, 25	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (ℤ≥‘𝑁) ∧ 𝑢 ∈ (ℕ0 ↑m (1...𝑘))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
27		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
28	27	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
29	26, 28	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ≥‘𝑁) ∧ 𝑢 ∈ (ℕ0 ↑m (1...𝑘))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
30	29	rexlimdva 3213	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
31	30	abssdv 4002	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ (ℕ0 ↑m (1...𝑁)))
32	15	elpw2 5269	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ 𝒫 (ℕ0 ↑m (1...𝑁)))
34		eleq1 2826	. . . . . . . . . . 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 246	. . . . . . . . . 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 3219	. . . . . . . . 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 3209	. . . . . . . 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 4003	. . . . . . 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 3955	. . . . . 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 5246	. . . . 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 6875	. . . 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 2824	. . 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 7308	. . . . . 6 ⊢ (ℕ0 ↑m (1...𝑘)) ∈ V
44	43	abrexex 7805	. . . . 5 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))𝑡 = (𝑢 ↾ (1...𝑁))} ∈ V
45		simpl 483	. . . . . . 7 ⊢ ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 = (𝑢 ↾ (1...𝑁)))
46	45	reximi 3178	. . . . . 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 4000	. . . . 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 5246	. . . 4 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ V
49	17, 48	elrnmpo 7410	. . 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 819	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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065   ⊆ wss 3887  𝒫 cpw 4533  dom cdm 5589  ran crn 5590   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615  0cc0 10871  1c1 10872  ℕ₀cn0 12233  ℤ_≥cuz 12582  ...cfz 13239  mzPolycmzp 40544  Diophcdioph 40577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-addcl 10931  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-dioph 40578

This theorem is referenced by:  eldioph  40580  eldioph2b  40585  eldiophelnn0  40586