eq0rabdioph - Mathbox for Stefan O'Rear

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Theorem eq0rabdioph 41499

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 1917	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑁 ∈ ℕ₀
2		nfmpt1 5255	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)
3	2	nfel1 2919	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
4	1, 3	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑡(𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5		zex 12563	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
6		nn0ssz 12577	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ⊆ ℤ
7		mapss 8879	. . . . . . . . . . . . . 14 ⊢ ((ℤ ∈ V ∧ ℕ₀ ⊆ ℤ) → (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁)))
8	5, 6, 7	mp2an 690	. . . . . . . . . . . . 13 ⊢ (ℕ₀ ↑_m (1...𝑁)) ⊆ (ℤ ↑_m (1...𝑁))
9	8	sseli 3977	. . . . . . . . . . . 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 41459	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴):(ℤ ↑_m (1...𝑁))⟶ℤ)
12		mptfcl 41443	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐴 ∈ ℤ)
15	14	adantll 712	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝐴 ∈ ℤ)
16		eqid 2732	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)
17	16	fvmpt2 7006	. . . . . . . . . . 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 2738	. . . . . . . . 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 2734	. . . . . . . 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 3254	. . . . . 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 3462	. . . . . 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 217	. . . . 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 2903	. . . . . 6 ⊢ Ⅎ𝑡(ℕ₀ ↑_m (1...𝑁))
26		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑎(ℕ₀ ↑_m (1...𝑁))
27		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑎((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28		nffvmpt1 6899	. . . . . . 7 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)‘𝑎)
29	28	nfeq1 2918	. . . . . 6 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30		fveqeq2 6897	. . . . . 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 3467	. . . . 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 2788	. . . 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 3433	. . . 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 2788	. . 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 8839	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ₀)
36		ffn 6714	. . . . . . . . . 10 ⊢ (𝑏:(1...𝑁)⟶ℕ₀ → 𝑏 Fn (1...𝑁))
37		fnresdm 6666	. . . . . . . . . 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 2743	. . . . . . . 8 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
40		equcom 2021	. . . . . . . 8 ⊢ (𝑎 = 𝑏 ↔ 𝑏 = 𝑎)
41	39, 40	bitrdi 286	. . . . . . 7 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
42	41	anbi1d 630	. . . . . 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 3092	. . . . 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 6897	. . . . . 6 ⊢ (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
45	44	ceqsrexbv 3643	. . . . 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 275	. . . 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 2802	. . 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 2788	. 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 12579	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
51		uzid 12833	. . . . 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 41481	. . 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 1371	. 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 2833	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑡 ∈ (ℤ ↑_m (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ 𝐴 = 0} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ⊆ wss 3947 ↦ cmpt 5230 ↾ cres 5677 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 0cc0 11106 1c1 11107 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 mzPolycmzp 41445 Diophcdioph 41478

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-mzpcl 41446 df-mzp 41447 df-dioph 41479

This theorem is referenced by: eqrabdioph 41500 0dioph 41501 vdioph 41502 rmydioph 41738 expdioph 41747

W3C validator