Theorem eq0rabdioph 37340

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 1843	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑁 ∈ ℕ0
2		nfmpt1 4747	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
3	2	nfel1 2779	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))
4	1, 3	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑡(𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
5		zex 11386	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
6		nn0ssz 11398	. . . . . . . . . . . . . 14 ⊢ ℕ0 ⊆ ℤ
7		mapss 7900	. . . . . . . . . . . . . 14 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁)))
8	5, 6, 7	mp2an 708	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑𝑚 (1...𝑁)) ⊆ (ℤ ↑𝑚 (1...𝑁))
9	8	sseli 3599	. . . . . . . . . . . 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 37299	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ)
12		mptfcl 37283	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) → 𝐴 ∈ ℤ))
13	12	imp 445	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴):(ℤ ↑𝑚 (1...𝑁))⟶ℤ ∧ 𝑡 ∈ (ℤ ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
14	11, 9, 13	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
15	14	adantll 750	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁))) → 𝐴 ∈ ℤ)
16		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) = (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)
17	16	fvmpt2 6291	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ∧ 𝐴 ∈ ℤ) → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 𝐴)
18	10, 15, 17	syl2anc 693	. . . . . . . . . 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 2628	. . . . . . . . 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 2624	. . . . . . . 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 450	. . . . . . 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 2957	. . . . . 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 3120	. . . . . 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 208	. . . . 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 2764	. . . . . 6 ⊢ Ⅎ𝑡(ℕ0 ↑𝑚 (1...𝑁))
26		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑎(ℕ0 ↑𝑚 (1...𝑁))
27		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑎((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0
28		nffvmpt1 6199	. . . . . . 7 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎)
29	28	nfeq1 2778	. . . . . 6 ⊢ Ⅎ𝑡((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0
30		fveq2 6191	. . . . . . 7 ⊢ (𝑡 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
31	30	eqeq1d 2624	. . . . . 6 ⊢ (𝑡 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑡) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
32	25, 26, 27, 29, 31	cbvrab 3198	. . . . 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}
33	24, 32	syl6eq 2672	. . . 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})
34		df-rab 2921	. . . 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)}
35	33, 34	syl6eq 2672	. . 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)})
36		elmapi 7879	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 (1...𝑁)) → 𝑏:(1...𝑁)⟶ℕ0)
37		ffn 6045	. . . . . . . . . 10 ⊢ (𝑏:(1...𝑁)⟶ℕ0 → 𝑏 Fn (1...𝑁))
38		fnresdm 6000	. . . . . . . . . 10 ⊢ (𝑏 Fn (1...𝑁) → (𝑏 ↾ (1...𝑁)) = 𝑏)
39	36, 37, 38	3syl 18	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 (1...𝑁)) → (𝑏 ↾ (1...𝑁)) = 𝑏)
40	39	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = 𝑏))
41		equcom 1945	. . . . . . . 8 ⊢ (𝑎 = 𝑏 ↔ 𝑏 = 𝑎)
42	40, 41	syl6bb 276	. . . . . . 7 ⊢ (𝑏 ∈ (ℕ0 ↑𝑚 (1...𝑁)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑏 = 𝑎))
43	42	anbi1d 741	. . . . . 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)))
44	43	rexbiia 3040	. . . . 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))
45		fveq2 6191	. . . . . . 7 ⊢ (𝑏 = 𝑎 → ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎))
46	45	eqeq1d 2624	. . . . . 6 ⊢ (𝑏 = 𝑎 → (((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑏) = 0 ↔ ((𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴)‘𝑎) = 0))
47	46	ceqsrexbv 3337	. . . . 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))
48	44, 47	bitr2i 265	. . . 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))
49	48	abbii 2739	. . 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)}
50	35, 49	syl6eq 2672	. 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)})
51		simpl 473	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ ℕ0)
52		nn0z 11400	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
53		uzid 11702	. . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
54	52, 53	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘𝑁))
55	54	adantr 481	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → 𝑁 ∈ (ℤ≥‘𝑁))
56		simpr 477	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁))) → (𝑡 ∈ (ℤ ↑𝑚 (1...𝑁)) ↦ 𝐴) ∈ (mzPoly‘(1...𝑁)))
57		eldioph 37321	. . 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))
58	51, 55, 56, 57	syl3anc 1326	. 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))
59	50, 58	eqeltrd 2701	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574   ↦ cmpt 4729   ↾ cres 5116   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  0cc0 9936  1c1 9937  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  mzPolycmzp 37285  Diophcdioph 37318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-mzpcl 37286  df-mzp 37287  df-dioph 37319

This theorem is referenced by:  eqrabdioph  37341  0dioph  37342  vdioph  37343  rmydioph  37581  expdioph  37590