Theorem eq0rabdioph 43621

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 1947	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑁 ∈ ℕ0
2		nfmpt1 5204	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
3	2	nfel1 2938	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
4	1, 3	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5		zex 12624	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
6		nn0ssz 12638	. . . . . . . . . . . . . 14 ⊢ ℕ0 ⊆ ℤ
7		mapss 8896	. . . . . . . . . . . . . 14 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁)))
8	5, 6, 7	mp2an 705	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁))
9	8	sseli 3927	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
10	9	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
11		mzpf 43581	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		mptfcl 43565	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) → 𝐴 ∈ ℤ))
13	12	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
14	11, 9, 13	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
15	14	adantll 727	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
16		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
17	16	fvmpt2 6998	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
18	10, 15, 17	syl2anc 596	. . . . . . . . . 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 2766	. . . . . . . . 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 2762	. . . . . . . 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 418	. . . . . . 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 3260	. . . . . 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 3441	. . . . . 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 221	. . . . 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 2922	. . . . . 6 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
26		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
27		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑎((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28		nffvmpt1 6889	. . . . . . 7 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎)
29	28	nfeq1 2937	. . . . . 6 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30		fveqeq2 6887	. . . . . 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 3446	. . . . 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 2811	. . . 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 3413	. . . 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 2811	. . 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 8848	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
36		ffn 6702	. . . . . . . . . 10 ⊢ (𝑏:(1...𝑁)⟶ℕ0 → 𝑏 Fn (1...𝑁))
37		fnresdm 6651	. . . . . . . . . 10 ⊢ (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
38	35, 36, 37	3syl 19	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
39	38	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
40		equcom 2051	. . . . . . . 8 ⊢ (𝑎 = 𝑏 ↔ 𝑏 = 𝑎)
41	39, 40	bitrdi 290	. . . . . . 7 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
42	41	anbi1d 643	. . . . . 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 3107	. . . . 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 6887	. . . . . 6 ⊢ (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
45	44	ceqsrexbv 3610	. . . . 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 279	. . . 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 2827	. . 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 2811	. 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 488	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
50		nn0z 12639	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
51		uzid 12902	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
52	50, 51	syl 18	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘𝑁))
53	52	adantr 486	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ≥‘𝑁))
54		simpr 490	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
55		eldioph 43603	. . 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 1398	. 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 2860	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899   ↦ cmpt 5186   ↾ cres 5657   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ↑_m cmap 8826  0cc0 11124  1c1 11125  ℕ₀cn0 12528  ℤcz 12615  ℤ_≥cuz 12887  ...cfz 13561  mzPolycmzp 43567  Diophcdioph 43600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-mzpcl 43568  df-mzp 43569  df-dioph 43601

This theorem is used by:  eqrabdioph  43622  0dioph  43623  vdioph  43624  rmydioph  43855  expdioph  43864