Theorem aannenlem1 25848

Description: Lemma for aannen 25851. (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 5152	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴))
2		breq2 5152	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
3	2	ralbidv 3177	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
4	1, 3	3anbi23d 1439	. . . . . 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 3440	. . . . 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 3326	. . . 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 3440	. . 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 11193	. . . 4 ⊢ ℂ ∈ V
10	9	rabex 5332	. . 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 6998	. 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 5055	. . 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 13939	. . . . . . 7 ⊢ (-𝐴...𝐴) ∈ Fin
14		fzfi 13939	. . . . . . 7 ⊢ (0...𝐴) ∈ Fin
15		mapfi 9350	. . . . . . 7 ⊢ (((-𝐴...𝐴) ∈ Fin ∧ (0...𝐴) ∈ Fin) → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
16	13, 14, 15	mp2an 690	. . . . . 6 ⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin
17	16	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
18		ovex 7444	. . . . . 6 ⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V
19		neeq1 3003	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝 ↔ 𝑎 ≠ 0𝑝))
20		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎))
21	20	breq1d 5158	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴))
22		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎))
23	22	fveq1d 6893	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒))
24	23	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒)))
25	24	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
26	25	ralbidv 3177	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
27	19, 21, 26	3anbi123d 1436	. . . . . . . . . 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 3683	. . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)
30	29	anim2i 617	. . . . . . . . 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 216	. . . . . . . 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 12571	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
33		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
34	33	coef2 25752	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
35	32, 34	mpan2 689	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Poly‘ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
36	35	ad2antrl 726	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶ℤ)
37	36	ffnd 6718	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎) Fn ℕ0)
38	35	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (coeff‘𝑎):ℕ0⟶ℤ)
39	38	ffvelcdmda 7086	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℤ)
40	39	zred 12668	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℝ)
41		nn0re 12483	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
42	41	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℝ)
43	40, 42	absled 15379	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
44		nn0z 12585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
45	44	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℤ)
46	45	znegcld 12670	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈ ℤ)
47		elfz 13492	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
50	49	biimpd 228	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
51	50	ralimdva 3167	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
52	51	impr 455	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))
53		fnfvrnss 7122	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝑎) Fn ℕ0 ∧ ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
54	37, 52, 53	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
55		df-f 6547	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)))
56	37, 54, 55	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶(-𝐴...𝐴))
57		fz0ssnn0 13598	. . . . . . . . . . 11 ⊢ (0...𝐴) ⊆ ℕ0
58		fssres 6757	. . . . . . . . . . 11 ⊢ (((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
59	56, 57, 58	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
60		ovex 7444	. . . . . . . . . . 11 ⊢ (-𝐴...𝐴) ∈ V
61		ovex 7444	. . . . . . . . . . 11 ⊢ (0...𝐴) ∈ V
62	60, 61	elmap 8867	. . . . . . . . . 10 ⊢ (((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
63	59, 62	sylibr 233	. . . . . . . . 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 413	. . . . . . . 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 1137	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴)
67	66	anim2i 617	. . . . . . . . 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 216	. . . . . . . 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 3003	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝 ↔ 𝑏 ≠ 0𝑝))
70		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏))
71	70	breq1d 5158	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴))
72		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏))
73	72	fveq1d 6893	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒))
74	73	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒)))
75	74	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
76	75	ralbidv 3177	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
77	69, 71, 76	3anbi123d 1436	. . . . . . . . . 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 3683	. . . . . . . . 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 1137	. . . . . . . . . 10 ⊢ ((𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴)
80	79	anim2i 617	. . . . . . . . 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 216	. . . . . . . 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 773	. . . . . . . . . . . . 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 25719	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Poly‘ℤ) → 𝑎:ℂ⟶ℂ)
84		ffn 6717	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 25719	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (Poly‘ℤ) → 𝑏:ℂ⟶ℂ)
88		ffn 6717	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 6893	. . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑))
94	93	adantl 482	. . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑))
96	95	adantl 482	. . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 15651	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 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 25754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ0)
103		nn0z 12585	. . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 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 12586	. . . . . . . . . . . . . . . 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 12838	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑎) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
108	104, 106, 107	syl2anc 584	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑎) = (deg‘𝑎)
112	33, 111	coeid3 25761	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 25754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (Poly‘ℤ) → (deg‘𝑏) ∈ ℕ0)
119		nn0z 12585	. . . . . . . . . . . . . . . . 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 12838	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑏) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
122	120, 106, 121	syl2anc 584	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . 15 ⊢ (coeff‘𝑏) = (coeff‘𝑏)
125		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑏) = (deg‘𝑏)
126	124, 125	coeid3 25761	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 7035	. . . . . . . . . . 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 457	. . . . . . . . . 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 6891	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏))
132	131	reseq1d 5980	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
133	130, 132	impbid1 224	. . . . . . . . 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 414	. . . . . . . 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 607	. . . . . . 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 8991	. . . . . 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 9194	. . . . 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 584	. . . 4 ⊢ (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
140		neeq1 3003	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝 ↔ 𝑐 ≠ 0𝑝))
141		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐))
142	141	breq1d 5158	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴))
143		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐))
144	143	fveq1d 6893	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒))
145	144	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒)))
146	145	breq1d 5158	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
147	146	ralbidv 3177	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
148	140, 142, 147	3anbi123d 1436	. . . . . . . 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 3683	. . . . . . 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 1136	. . . . . . . 8 ⊢ ((𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝)
151	150	anim2i 617	. . . . . . 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 216	. . . . . 6 ⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
153		fveqeq2 6900	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑎 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑎) = 0))
154	153	elrab 3683	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0))
155		plyf 25719	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐:ℂ⟶ℂ)
156	155	ffnd 6718	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐 Fn ℂ)
157	156	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → 𝑐 Fn ℂ)
158		fniniseg 7061	. . . . . . . . . . 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 289	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ 𝑎 ∈ (◡𝑐 “ {0})))
161	160	eqrdv 2730	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = (◡𝑐 “ {0}))
162		eqid 2732	. . . . . . . . . 10 ⊢ (◡𝑐 “ {0}) = (◡𝑐 “ {0})
163	162	fta1 25828	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → ((◡𝑐 “ {0}) ∈ Fin ∧ (♯‘(◡𝑐 “ {0})) ≤ (deg‘𝑐)))
164	163	simpld 495	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (◡𝑐 “ {0}) ∈ Fin)
165	161, 164	eqeltrd 2833	. . . . . . 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 3145	. . . 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 9342	. . . 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 584	. . 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 2838	. 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 2833	1 ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ⊆ wss 3948  {csn 4628  ∪ ciun 4997   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5675  ran crn 5677   ↾ cres 5678   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411   ↑_m cmap 8822   ≼ cdom 8939  Fincfn 8941  ℂcc 11110  ℝcr 11111  0cc0 11112   · cmul 11117   ≤ cle 11251  -cneg 11447  ℕ₀cn0 12474  ℤcz 12560  ℤ_≥cuz 12824  ...cfz 13486  ↑cexp 14029  ♯chash 14292  abscabs 15183  Σcsu 15634  0_𝑝c0p 25193  Polycply 25705  coeffccoe 25707  degcdgr 25708

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-fzo 13630  df-fl 13759  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-rlim 15435  df-sum 15635  df-0p 25194  df-ply 25709  df-idp 25710  df-coe 25711  df-dgr 25712  df-quot 25811

This theorem is referenced by:  aannenlem3  25850