Theorem eldiophb 43209

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 43208	. . . 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 6201	. . 3 ⊢ dom Dioph ⊆ ℕ0
3		elfvdm 6870	. . 3 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ dom Dioph)
4	2, 3	sselid 3920	. 2 ⊢ (𝐷 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
5		fveq2 6836	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (ℤ≥‘𝑛) = (ℤ≥‘𝑁))
6		eqidd 2738	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝑘)))
7		oveq2 7370	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
8	7	reseq2d 5940	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (𝑢 ↾ (1...𝑛)) = (𝑢 ↾ (1...𝑁)))
9	8	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑡 = (𝑢 ↾ (1...𝑛)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
10	9	anbi1d 632	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)))
11	10	rexbidv 3162	. . . . . . . 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 2803	. . . . . . 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 7437	. . . . . 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 5889	. . . . 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 7395	. . . . . . 7 ⊢ (ℕ0 ↑m (1...𝑁)) ∈ V
16	15	pwex 5319	. . . . . 6 ⊢ 𝒫 (ℕ0 ↑m (1...𝑁)) ∈ V
17		eqid 2737	. . . . . . . 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 7495	. . . . . . 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 8791	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑m (1...𝑘)) → 𝑢:(1...𝑘)⟶ℕ0)
20		fzss2 13513	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑘))
21		fssres 6702	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢:(1...𝑘)⟶ℕ0 ∧ (1...𝑁) ⊆ (1...𝑘)) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
22	19, 20, 21	syl2anr 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (ℤ≥‘𝑁) ∧ 𝑢 ∈ (ℕ0 ↑m (1...𝑘))) → (𝑢 ↾ (1...𝑁)):(1...𝑁)⟶ℕ0)
23		nn0ex 12438	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
24		ovex 7395	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
25	23, 24	elmap 8814	. . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → (∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → 𝑡 ∈ (ℕ0 ↑m (1...𝑁))))
31	30	abssdv 4008	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑁) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ⊆ (ℕ0 ↑m (1...𝑁)))
32	15	elpw2 5272	. . . . . . . . . . . 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 2825	. . . . . . . . . . 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 3144	. . . . . . . . 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 3132	. . . . . . . 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 4009	. . . . . . 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 3969	. . . . . 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 5260	. . . . 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 6943	. . . 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 2823	. . 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 7395	. . . . . 6 ⊢ (ℕ0 ↑m (1...𝑘)) ∈ V
44	43	abrexex 7910	. . . . 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 3076	. . . . . 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 4007	. . . . 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 5260	. . . 4 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ V
49	17, 48	elrnmpo 7498	. . 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 822	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 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ⊆ wss 3890  𝒫 cpw 4542  dom cdm 5626  ran crn 5627   ↾ cres 5628  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∈ cmpo 7364   ↑_m cmap 8768  0cc0 11033  1c1 11034  ℕ₀cn0 12432  ℤ_≥cuz 12783  ...cfz 13456  mzPolycmzp 43174  Diophcdioph 43207

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-addcl 11093  ax-pre-lttri 11107  ax-pre-lttrn 11108

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 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-neg 11375  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-dioph 43208

This theorem is referenced by:  eldioph  43210  eldioph2b  43215  eldiophelnn0  43216