Theorem aannenlem1 23887

Description: Lemma for aannen 23890. (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 4587	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴))
2		breq2 4587	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
3	2	ralbidv 2969	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
4	1, 3	3anbi23d 1394	. . . . . 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 3164	. . . . 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 3122	. . . 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 3164	. . 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 9896	. . . 4 ⊢ ℂ ∈ V
10	9	rabex 4740	. . 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 6191	. 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 4503	. . 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 12633	. . . . . . 7 ⊢ (-𝐴...𝐴) ∈ Fin
14		fzfi 12633	. . . . . . 7 ⊢ (0...𝐴) ∈ Fin
15		mapfi 8145	. . . . . . 7 ⊢ (((-𝐴...𝐴) ∈ Fin ∧ (0...𝐴) ∈ Fin) → ((-𝐴...𝐴) ↑𝑚 (0...𝐴)) ∈ Fin)
16	13, 14, 15	mp2an 704	. . . . . 6 ⊢ ((-𝐴...𝐴) ↑𝑚 (0...𝐴)) ∈ Fin
17	16	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → ((-𝐴...𝐴) ↑𝑚 (0...𝐴)) ∈ Fin)
18		ovex 6577	. . . . . 6 ⊢ ((-𝐴...𝐴) ↑𝑚 (0...𝐴)) ∈ V
19		neeq1 2844	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝 ↔ 𝑎 ≠ 0𝑝))
20		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎))
21	20	breq1d 4593	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴))
22		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎))
23	22	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒))
24	23	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒)))
25	24	breq1d 4593	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
26	25	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
27	19, 21, 26	3anbi123d 1391	. . . . . . . . . 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 3331	. . . . . . . . 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 1056	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)
30	29	anim2i 591	. . . . . . . . 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 206	. . . . . . . 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 11265	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
33		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
34	33	coef2 23791	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
35	32, 34	mpan2 703	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Poly‘ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
36	35	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶ℤ)
37	36	ffnd 5959	. . . . . . . . . . . 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	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℤ)
40	39	zred 11358	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℝ)
41		nn0re 11178	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
42	41	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℝ)
43	40, 42	absled 14017	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
44		nn0z 11277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
45	44	ad2antrr 758	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℤ)
46	45	znegcld 11360	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈ ℤ)
47		elfz 12203	. . . . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
50	49	biimpd 218	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
51	50	ralimdva 2945	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
52	51	impr 647	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))
53		fnfvrnss 6297	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝑎) Fn ℕ0 ∧ ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
54	37, 52, 53	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
55		df-f 5808	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)))
56	37, 54, 55	sylanbrc 695	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶(-𝐴...𝐴))
57		fz0ssnn0 12304	. . . . . . . . . . 11 ⊢ (0...𝐴) ⊆ ℕ0
58		fssres 5983	. . . . . . . . . . 11 ⊢ (((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
59	56, 57, 58	sylancl 693	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
60		ovex 6577	. . . . . . . . . . 11 ⊢ (-𝐴...𝐴) ∈ V
61		ovex 6577	. . . . . . . . . . 11 ⊢ (0...𝐴) ∈ V
62	60, 61	elmap 7772	. . . . . . . . . 10 ⊢ (((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑𝑚 (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
63	59, 62	sylibr 223	. . . . . . . . 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 449	. . . . . . . 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 33	. . . . . . 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 1055	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴)
67	66	anim2i 591	. . . . . . . . 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 206	. . . . . . . 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 2844	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝 ↔ 𝑏 ≠ 0𝑝))
70		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏))
71	70	breq1d 4593	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴))
72		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏))
73	72	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒))
74	73	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒)))
75	74	breq1d 4593	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
76	75	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
77	69, 71, 76	3anbi123d 1391	. . . . . . . . . 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 3331	. . . . . . . . 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 1055	. . . . . . . . . 10 ⊢ ((𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴)
80	79	anim2i 591	. . . . . . . . 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 206	. . . . . . . 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 794	. . . . . . . . . . . . 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 23758	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Poly‘ℤ) → 𝑎:ℂ⟶ℂ)
84		ffn 5958	. . . . . . . . . . . . 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 796	. . . . . . . . . . . . 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 23758	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (Poly‘ℤ) → 𝑏:ℂ⟶ℂ)
88		ffn 5958	. . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . . . 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 6105	. . . . . . . . . . . . . . . 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 6117	. . . . . . . . . . . . . . . . 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 6117	. . . . . . . . . . . . . . . . 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 2652	. . . . . . . . . . . . . . 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 6564	. . . . . . . . . . . . . 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 14281	. . . . . . . . . . . . 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 802	. . . . . . . . . . . . . 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 803	. . . . . . . . . . . . . . 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 23793	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ0)
103		nn0z 11277	. . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . 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 11356	. . . . . . . . . . . . . . . 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 11577	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑎) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
108	104, 106, 107	syl2anc 691	. . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑎) = (deg‘𝑎)
112	33, 111	coeid3 23800	. . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . 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 1119	. . . . . . . . . . . . . . 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 1257	. . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . 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 23793	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (Poly‘ℤ) → (deg‘𝑏) ∈ ℕ0)
119		nn0z 11277	. . . . . . . . . . . . . . . . 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 11577	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑏) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
122	120, 106, 121	syl2anc 691	. . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . 15 ⊢ (coeff‘𝑏) = (coeff‘𝑏)
125		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑏) = (deg‘𝑏)
126	124, 125	coeid3 23800	. . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . 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 2654	. . . . . . . . . . . 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 6222	. . . . . . . . . . 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 641	. . . . . . . . . 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 6103	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏))
132	131	reseq1d 5316	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
133	130, 132	impbid1 214	. . . . . . . . 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 450	. . . . . . . 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 686	. . . . . . 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 7882	. . . . . 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 8066	. . . . 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 691	. . . 4 ⊢ (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
140		neeq1 2844	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝 ↔ 𝑐 ≠ 0𝑝))
141		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐))
142	141	breq1d 4593	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴))
143		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐))
144	143	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒))
145	144	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒)))
146	145	breq1d 4593	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
147	146	ralbidv 2969	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
148	140, 142, 147	3anbi123d 1391	. . . . . . . 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 3331	. . . . . . 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 1054	. . . . . . . 8 ⊢ ((𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝)
151	150	anim2i 591	. . . . . . 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 206	. . . . . 6 ⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
153		plyf 23758	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐:ℂ⟶ℂ)
154	153	ffnd 5959	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐 Fn ℂ)
155	154	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → 𝑐 Fn ℂ)
156		fniniseg 6246	. . . . . . . . . . 11 ⊢ (𝑐 Fn ℂ → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)))
157	155, 156	syl 17	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)))
158		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑎 → (𝑐‘𝑏) = (𝑐‘𝑎))
159	158	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑎 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑎) = 0))
160	159	elrab 3331	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0))
161	157, 160	syl6rbbr 278	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ 𝑎 ∈ (◡𝑐 “ {0})))
162	161	eqrdv 2608	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = (◡𝑐 “ {0}))
163		eqid 2610	. . . . . . . . . 10 ⊢ (◡𝑐 “ {0}) = (◡𝑐 “ {0})
164	163	fta1 23867	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → ((◡𝑐 “ {0}) ∈ Fin ∧ (#‘(◡𝑐 “ {0})) ≤ (deg‘𝑐)))
165	164	simpld 474	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (◡𝑐 “ {0}) ∈ Fin)
166	162, 165	eqeltrd 2688	. . . . . . 7 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)
167	166	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0 → ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin))
168	152, 167	syl5 33	. . . . 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))
169	168	ralrimiv 2948	. . . 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)
170		iunfi 8137	. . . 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)
171	139, 169, 170	syl2anc 691	. . 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)
172	12, 171	syl5eqelr 2693	. 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)
173	11, 172	eqeltrd 2688	1 ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   ≼ cdom 7839  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815   · cmul 9820   ≤ cle 9954  -cneg 10146  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ↑cexp 12722  #chash 12979  abscabs 13822  Σcsu 14264  0_𝑝c0p 23242  Polycply 23744  coeffccoe 23746  degcdgr 23747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-idp 23749  df-coe 23750  df-dgr 23751  df-quot 23850

This theorem is referenced by:  aannenlem3  23889