Theorem aannenlem1 26296

Description: Lemma for aannen 26299. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Hypothesis

Ref	Expression
aannenlem.a	⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})

Assertion

Ref	Expression
aannenlem1	⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin)

Distinct variable group:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒

Allowed substitution hints:   𝐻(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aannenlem1

Step	Hyp	Ref	Expression
1		breq2 5103	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴))
2		breq2 5103	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
3	2	ralbidv 3160	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
4	1, 3	3anbi23d 1442	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)))
5	4	rabbidv 3407	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} = {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)})
6	5	rexeqdv 3298	. . . 4 ⊢ (𝑎 = 𝐴 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0))
7	6	rabbidv 3407	. . 3 ⊢ (𝑎 = 𝐴 → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0})
8		aannenlem.a	. . 3 ⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})
9		cnex 11111	. . . 4 ⊢ ℂ ∈ V
10	9	rabex 5285	. . 3 ⊢ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0} ∈ V
11	7, 8, 10	fvmpt 6942	. 2 ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0})
12		iunrab 5009	. . 3 ⊢ ∪ 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0}
13		fzfi 13899	. . . . . . 7 ⊢ (-𝐴...𝐴) ∈ Fin
14		fzfi 13899	. . . . . . 7 ⊢ (0...𝐴) ∈ Fin
15		mapfi 9252	. . . . . . 7 ⊢ (((-𝐴...𝐴) ∈ Fin ∧ (0...𝐴) ∈ Fin) → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
16	13, 14, 15	mp2an 693	. . . . . 6 ⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin
17	16	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
18		ovex 7393	. . . . . 6 ⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V
19		neeq1 2995	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝 ↔ 𝑎 ≠ 0𝑝))
20		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎))
21	20	breq1d 5109	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴))
22		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎))
23	22	fveq1d 6837	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒))
24	23	fveq2d 6839	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒)))
25	24	breq1d 5109	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
26	25	ralbidv 3160	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
27	19, 21, 26	3anbi123d 1439	. . . . . . . . . 10 ⊢ (𝑑 = 𝑎 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)))
28	27	elrab 3647	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)))
29		simp3 1139	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)
30	29	anim2i 618	. . . . . . . . 9 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
31	28, 30	sylbi 217	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
32		0z 12503	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
33		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
34	33	coef2 26196	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
35	32, 34	mpan2 692	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Poly‘ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
36	35	ad2antrl 729	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶ℤ)
37	36	ffnd 6664	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎) Fn ℕ0)
38	35	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (coeff‘𝑎):ℕ0⟶ℤ)
39	38	ffvelcdmda 7031	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℤ)
40	39	zred 12600	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℝ)
41		nn0re 12414	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
42	41	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℝ)
43	40, 42	absled 15360	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
44		nn0z 12516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
45	44	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℤ)
46	45	znegcld 12602	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈ ℤ)
47		elfz 13433	. . . . . . . . . . . . . . . . . 18 ⊢ ((((coeff‘𝑎)‘𝑒) ∈ ℤ ∧ -𝐴 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴) ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
48	39, 46, 45, 47	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → (((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴) ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
49	43, 48	bitr4d 282	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
50	49	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
51	50	ralimdva 3149	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
52	51	impr 454	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))
53		fnfvrnss 7068	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝑎) Fn ℕ0 ∧ ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
54	37, 52, 53	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
55		df-f 6497	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)))
56	37, 54, 55	sylanbrc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶(-𝐴...𝐴))
57		fz0ssnn0 13542	. . . . . . . . . . 11 ⊢ (0...𝐴) ⊆ ℕ0
58		fssres 6701	. . . . . . . . . . 11 ⊢ (((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
59	56, 57, 58	sylancl 587	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
60		ovex 7393	. . . . . . . . . . 11 ⊢ (-𝐴...𝐴) ∈ V
61		ovex 7393	. . . . . . . . . . 11 ⊢ (0...𝐴) ∈ V
62	60, 61	elmap 8813	. . . . . . . . . 10 ⊢ (((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
63	59, 62	sylibr 234	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)))
64	63	ex 412	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → ((𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴))))
65	31, 64	syl5 34	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴))))
66		simp2 1138	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴)
67	66	anim2i 618	. . . . . . . . 9 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴))
68	28, 67	sylbi 217	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴))
69		neeq1 2995	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝 ↔ 𝑏 ≠ 0𝑝))
70		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏))
71	70	breq1d 5109	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴))
72		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏))
73	72	fveq1d 6837	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒))
74	73	fveq2d 6839	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒)))
75	74	breq1d 5109	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
76	75	ralbidv 3160	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
77	69, 71, 76	3anbi123d 1439	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)))
78	77	elrab 3647	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑏 ∈ (Poly‘ℤ) ∧ (𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)))
79		simp2 1138	. . . . . . . . . 10 ⊢ ((𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴)
80	79	anim2i 618	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Poly‘ℤ) ∧ (𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)) → (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴))
81	78, 80	sylbi 217	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴))
82		simplll 775	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑎 ∈ (Poly‘ℤ))
83		plyf 26163	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Poly‘ℤ) → 𝑎:ℂ⟶ℂ)
84		ffn 6663	. . . . . . . . . . . . 13 ⊢ (𝑎:ℂ⟶ℂ → 𝑎 Fn ℂ)
85	82, 83, 84	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑎 Fn ℂ)
86		simplrl 777	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑏 ∈ (Poly‘ℤ))
87		plyf 26163	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (Poly‘ℤ) → 𝑏:ℂ⟶ℂ)
88		ffn 6663	. . . . . . . . . . . . 13 ⊢ (𝑏:ℂ⟶ℂ → 𝑏 Fn ℂ)
89	86, 87, 88	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑏 Fn ℂ)
90		simplrr 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
91	90	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
92	91	fveq1d 6837	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑))
93		fvres 6854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑))
94	93	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑))
95		fvres 6854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑))
96	95	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑))
97	92, 94, 96	3eqtr3d 2780	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → ((coeff‘𝑎)‘𝑑) = ((coeff‘𝑏)‘𝑑))
98	97	oveq1d 7375	. . . . . . . . . . . . . 14 ⊢ ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑)) = (((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑)))
99	98	sumeq2dv 15629	. . . . . . . . . . . . 13 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑)) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑)))
100		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑎 ∈ (Poly‘ℤ))
101		simp-4r 784	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑎) ≤ 𝐴)
102		dgrcl 26198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ0)
103		nn0z 12516	. . . . . . . . . . . . . . . . 17 ⊢ ((deg‘𝑎) ∈ ℕ0 → (deg‘𝑎) ∈ ℤ)
104	100, 102, 103	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑎) ∈ ℤ)
105		simplrl 777	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ ℕ0)
106	105	nn0zd 12517	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ ℤ)
107		eluz 12769	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑎) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
108	104, 106, 107	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
109	101, 108	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ (ℤ≥‘(deg‘𝑎)))
110		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
111		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑎) = (deg‘𝑎)
112	33, 111	coeid3 26205	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ (ℤ≥‘(deg‘𝑎)) ∧ 𝑐 ∈ ℂ) → (𝑎‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑)))
113	100, 109, 110, 112	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑎‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑)))
114		simp1rl 1240	. . . . . . . . . . . . . . 15 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴))) ∧ 𝑐 ∈ ℂ) → 𝑏 ∈ (Poly‘ℤ))
115	114	3expa 1119	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑏 ∈ (Poly‘ℤ))
116		simplrr 778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → (deg‘𝑏) ≤ 𝐴)
117	116	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑏) ≤ 𝐴)
118		dgrcl 26198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (Poly‘ℤ) → (deg‘𝑏) ∈ ℕ0)
119		nn0z 12516	. . . . . . . . . . . . . . . . 17 ⊢ ((deg‘𝑏) ∈ ℕ0 → (deg‘𝑏) ∈ ℤ)
120	115, 118, 119	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑏) ∈ ℤ)
121		eluz 12769	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑏) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
122	120, 106, 121	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
123	117, 122	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ (ℤ≥‘(deg‘𝑏)))
124		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (coeff‘𝑏) = (coeff‘𝑏)
125		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑏) = (deg‘𝑏)
126	124, 125	coeid3 26205	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ (ℤ≥‘(deg‘𝑏)) ∧ 𝑐 ∈ ℂ) → (𝑏‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑)))
127	115, 123, 110, 126	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑏‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑)))
128	99, 113, 127	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑎‘𝑐) = (𝑏‘𝑐))
129	85, 89, 128	eqfnfvd 6981	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑎 = 𝑏)
130	129	expr 456	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ 𝐴 ∈ ℕ0) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) → 𝑎 = 𝑏))
131		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏))
132	131	reseq1d 5938	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
133	130, 132	impbid1 225	. . . . . . . . 9 ⊢ ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ 𝐴 ∈ ℕ0) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏))
134	133	expcom 413	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → (((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏)))
135	68, 81, 134	syl2ani 608	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → ((𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∧ 𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)}) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏)))
136	65, 135	dom2d 8934	. . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴))))
137	18, 136	mpi 20	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴)))
138		domfi 9117	. . . . 5 ⊢ ((((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin ∧ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴))) → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
139	17, 137, 138	syl2anc 585	. . . 4 ⊢ (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
140		neeq1 2995	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝 ↔ 𝑐 ≠ 0𝑝))
141		fveq2 6835	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐))
142	141	breq1d 5109	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴))
143		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐))
144	143	fveq1d 6837	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒))
145	144	fveq2d 6839	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒)))
146	145	breq1d 5109	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
147	146	ralbidv 3160	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
148	140, 142, 147	3anbi123d 1439	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)))
149	148	elrab 3647	. . . . . . 7 ⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑐 ∈ (Poly‘ℤ) ∧ (𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)))
150		simp1 1137	. . . . . . . 8 ⊢ ((𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝)
151	150	anim2i 618	. . . . . . 7 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ (𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)) → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
152	149, 151	sylbi 217	. . . . . 6 ⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
153		fveqeq2 6844	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑎 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑎) = 0))
154	153	elrab 3647	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0))
155		plyf 26163	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐:ℂ⟶ℂ)
156	155	ffnd 6664	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐 Fn ℂ)
157	156	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → 𝑐 Fn ℂ)
158		fniniseg 7007	. . . . . . . . . . 11 ⊢ (𝑐 Fn ℂ → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)))
159	157, 158	syl 17	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)))
160	154, 159	bitr4id 290	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ 𝑎 ∈ (◡𝑐 “ {0})))
161	160	eqrdv 2735	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = (◡𝑐 “ {0}))
162		eqid 2737	. . . . . . . . . 10 ⊢ (◡𝑐 “ {0}) = (◡𝑐 “ {0})
163	162	fta1 26276	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → ((◡𝑐 “ {0}) ∈ Fin ∧ (♯‘(◡𝑐 “ {0})) ≤ (deg‘𝑐)))
164	163	simpld 494	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (◡𝑐 “ {0}) ∈ Fin)
165	161, 164	eqeltrd 2837	. . . . . . 7 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)
166	165	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0 → ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin))
167	152, 166	syl5 34	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin))
168	167	ralrimiv 3128	. . . 4 ⊢ (𝐴 ∈ ℕ0 → ∀𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)
169		iunfi 9247	. . . 4 ⊢ (({𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin ∧ ∀𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin) → ∪ 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)
170	139, 168, 169	syl2anc 585	. . 3 ⊢ (𝐴 ∈ ℕ0 → ∪ 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)
171	12, 170	eqeltrrid 2842	. 2 ⊢ (𝐴 ∈ ℕ0 → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0} ∈ Fin)
172	11, 171	eqeltrd 2837	1 ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400  Vcvv 3441   ⊆ wss 3902  {csn 4581  ∪ ciun 4947   class class class wbr 5099   ↦ cmpt 5180  ^◡ccnv 5624  ran crn 5626   ↾ cres 5627   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ↑_m cmap 8767   ≼ cdom 8885  Fincfn 8887  ℂcc 11028  ℝcr 11029  0cc0 11030   · cmul 11035   ≤ cle 11171  -cneg 11369  ℕ₀cn0 12405  ℤcz 12492  ℤ_≥cuz 12755  ...cfz 13427  ↑cexp 13988  ♯chash 14257  abscabs 15161  Σcsu 15613  0_𝑝c0p 25630  Polycply 26149  coeffccoe 26151  degcdgr 26152

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-oi 9419  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-xnn0 12479  df-z 12493  df-uz 12756  df-rp 12910  df-fz 13428  df-fzo 13575  df-fl 13716  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-clim 15415  df-rlim 15416  df-sum 15614  df-0p 25631  df-ply 26153  df-idp 26154  df-coe 26155  df-dgr 26156  df-quot 26259

This theorem is referenced by:  aannenlem3  26298