Theorem eq0rabdioph 43225

Description: This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Assertion

Ref	Expression
eq0rabdioph	⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))

Distinct variable group:   𝑡,𝑁

Allowed substitution hint:   𝐴(𝑡)

Proof of Theorem eq0rabdioph

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑁 ∈ ℕ0
2		nfmpt1 5171	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
3	2	nfel1 2917	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
4	1, 3	nfan 1906	. . . . . . 7 ⊢ Ⅎ𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5		zex 12524	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
6		nn0ssz 12538	. . . . . . . . . . . . . 14 ⊢ ℕ0 ⊆ ℤ
7		mapss 8827	. . . . . . . . . . . . . 14 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁)))
8	5, 6, 7	mp2an 698	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁))
9	8	sseli 3911	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
10	9	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
11		mzpf 43185	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		mptfcl 43169	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) → 𝐴 ∈ ℤ))
13	12	imp 407	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
14	11, 9, 13	syl2an 602	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
15	14	adantll 720	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
16		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
17	16	fvmpt2 6947	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
18	10, 15, 17	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
19	18	eqcomd 2745	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 = ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡))
20	19	eqeq1d 2741	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → (𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0))
21	20	ex 413	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) → (𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0)))
22	4, 21	ralrimi 3237	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → ∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0))
23		rabbi 3421	. . . . . 6 ⊢ (∀𝑡 ∈ (ℕ0 ↑m (1...𝑁))(𝐴 = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0) ↔ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0})
24	22, 23	sylib 219	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0})
25		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
26		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
27		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑎((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28		nffvmpt1 6838	. . . . . . 7 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎)
29	28	nfeq1 2916	. . . . . 6 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30		fveqeq2 6836	. . . . . 6 ⊢ (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
31	25, 26, 27, 29, 30	cbvrabw 3426	. . . . 5 ⊢ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0} = {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0}
32	24, 31	eqtrdi 2790	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} = {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0})
33		df-rab 3392	. . . 4 ⊢ {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0} = {𝑎 ∣ (𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0)}
34	32, 33	eqtrdi 2790	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} = {𝑎 ∣ (𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0)})
35		elmapi 8786	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
36		ffn 6655	. . . . . . . . . 10 ⊢ (𝑏:(1...𝑁)⟶ℕ0 → 𝑏 Fn (1...𝑁))
37		fnresdm 6604	. . . . . . . . . 10 ⊢ (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
38	35, 36, 37	3syl 18	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
39	38	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
40		equcom 2025	. . . . . . . 8 ⊢ (𝑎 = 𝑏 ↔ 𝑏 = 𝑎)
41	39, 40	bitrdi 288	. . . . . . 7 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
42	41	anbi1d 637	. . . . . 6 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0) ↔ (𝑏 = 𝑎 ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)))
43	42	rexbiia 3084	. . . . 5 ⊢ (∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑏 = 𝑎 ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0))
44		fveqeq2 6836	. . . . . 6 ⊢ (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
45	44	ceqsrexbv 3594	. . . . 5 ⊢ (∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑏 = 𝑎 ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0) ↔ (𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
46	43, 45	bitr2i 277	. . . 4 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0) ↔ ∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0))
47	46	abbii 2806	. . 3 ⊢ {𝑎 ∣ (𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)}
48	34, 47	eqtrdi 2790	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)})
49		simpl 483	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
50		nn0z 12539	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
51		uzid 12794	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
52	50, 51	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘𝑁))
53	52	adantr 481	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ≥‘𝑁))
54		simpr 485	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
55		eldioph 43207	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑁) ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)} ∈ (Dioph‘𝑁))
56	49, 53, 54, 55	syl3anc 1379	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑎 ∣ ∃𝑏 ∈ (ℕ0 ↑m (1...𝑁))(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0)} ∈ (Dioph‘𝑁))
57	48, 56	eqeltrd 2839	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431   ⊆ wss 3883   ↦ cmpt 5153   ↾ cres 5620   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  0cc0 11029  1c1 11030  ℕ₀cn0 12428  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  mzPolycmzp 43171  Diophcdioph 43204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-mzpcl 43172  df-mzp 43173  df-dioph 43205

This theorem is referenced by:  eqrabdioph  43226  0dioph  43227  vdioph  43228  rmydioph  43459  expdioph  43468