Theorem eq0rabdioph 43018

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 1915	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑁 ∈ ℕ0
2		nfmpt1 5197	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
3	2	nfel1 2915	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
4	1, 3	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5		zex 12497	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
6		nn0ssz 12511	. . . . . . . . . . . . . 14 ⊢ ℕ0 ⊆ ℤ
7		mapss 8827	. . . . . . . . . . . . . 14 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁)))
8	5, 6, 7	mp2an 692	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m (1...𝑁)) ⊆ (ℤ ↑m (1...𝑁))
9	8	sseli 3929	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
10	9	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝑡 ∈ (ℤ ↑m (1...𝑁)))
11		mzpf 42978	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ)
12		mptfcl 42962	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑m (1...𝑁)) → 𝐴 ∈ ℤ))
13	12	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴):(ℤ ↑m (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
14	11, 9, 13	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
15	14	adantll 714	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → 𝐴 ∈ ℤ)
16		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)
17	16	fvmpt2 6952	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
18	10, 15, 17	syl2anc 584	. . . . . . . . . 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 2742	. . . . . . . . 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 2738	. . . . . . . 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 412	. . . . . . 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 3234	. . . . . 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 3429	. . . . . 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 218	. . . . 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 2898	. . . . . 6 ⊢ Ⅎ𝑡(ℕ0 ↑m (1...𝑁))
26		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
27		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑎((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28		nffvmpt1 6845	. . . . . . 7 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎)
29	28	nfeq1 2914	. . . . . 6 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30		fveqeq2 6843	. . . . . 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 3434	. . . . 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 2787	. . . 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 3400	. . . 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 2787	. . 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 6662	. . . . . . . . . 10 ⊢ (𝑏:(1...𝑁)⟶ℕ0 → 𝑏 Fn (1...𝑁))
37		fnresdm 6611	. . . . . . . . . 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 2747	. . . . . . . 8 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
40		equcom 2019	. . . . . . . 8 ⊢ (𝑎 = 𝑏 ↔ 𝑏 = 𝑎)
41	39, 40	bitrdi 287	. . . . . . 7 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
42	41	anbi1d 631	. . . . . 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 3081	. . . . 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 6843	. . . . . 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 276	. . . 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 2803	. . 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 2787	. 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 482	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
50		nn0z 12512	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
51		uzid 12766	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
52	50, 51	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘𝑁))
53	52	adantr 480	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ≥‘𝑁))
54		simpr 484	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
55		eldioph 43000	. . 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 1373	. 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 2836	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901   ↦ cmpt 5179   ↾ cres 5626   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  0cc0 11026  1c1 11027  ℕ₀cn0 12401  ℤcz 12488  ℤ_≥cuz 12751  ...cfz 13423  mzPolycmzp 42964  Diophcdioph 42997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-mzpcl 42965  df-mzp 42966  df-dioph 42998

This theorem is referenced by:  eqrabdioph  43019  0dioph  43020  vdioph  43021  rmydioph  43256  expdioph  43265