Theorem aannenlem1 26682

Description: Lemma for aannen 26685. (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 5110	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴))
2		breq2 5110	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
3	2	ralbidv 3189	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
4	1, 3	3anbi23d 1468	. . . . . 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 3423	. . . . 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 3324	. . . 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 3423	. . 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 11305	. . . 4 ⊢ ℂ ∈ V
10	9	rabex 5304	. . 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 7013	. 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 5014	. . 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 14139	. . . . . . 7 ⊢ ( -𝐴...𝐴) ∈ Fin
14		fzfi 14139	. . . . . . 7 ⊢ (0...𝐴) ∈ Fin
15		mapfi 9362	. . . . . . 7 ⊢ ((( -𝐴...𝐴) ∈ Fin ∧ (0...𝐴) ∈ Fin) → (( -𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
16	13, 14, 15	mp2an 705	. . . . . 6 ⊢ (( -𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin
17	16	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℕ0 → (( -𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
18		ovex 7473	. . . . . 6 ⊢ (( -𝐴...𝐴) ↑m (0...𝐴)) ∈ V
19		neeq1 3021	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝 ↔ 𝑎 ≠ 0𝑝))
20		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎))
21	20	breq1d 5116	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴))
22		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎))
23	22	fveq1d 6907	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒))
24	23	fveq2d 6909	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒)))
25	24	breq1d 5116	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
26	25	ralbidv 3189	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑎 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
27	19, 21, 26	3anbi123d 1465	. . . . . . . . . 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 3648	. . . . . . . . 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 1156	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)
30	29	anim2i 629	. . . . . . . . 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 220	. . . . . . . 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 12728	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
33		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (coeff‘𝑎) = (coeff‘𝑎)
34	33	coef2 26577	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
35	32, 34	mpan2 704	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (Poly‘ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
36	35	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶ℤ)
37	36	ffnd 6729	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎) Fn ℕ0)
38	35	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (coeff‘𝑎):ℕ0⟶ℤ)
39	38	ffvelcdmda 7105	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℤ)
40	39	zred 12827	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℝ)
41		nn0re 12639	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
42	41	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℝ)
43	40, 42	absled 15626	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ( -𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
44		nn0z 12741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ)
45	44	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℤ)
46	45	znegcld 12829	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈ ℤ)
47		elfz 13669	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 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 285	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ ( -𝐴...𝐴)))
50	49	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ ( -𝐴...𝐴)))
51	50	ralimdva 3178	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑎 ∈ (Poly‘ℤ)) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ ( -𝐴...𝐴)))
52	51	impr 460	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ ( -𝐴...𝐴))
53		fnfvrnss 7142	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝑎) Fn ℕ0 ∧ ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ ( -𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ ( -𝐴...𝐴))
54	37, 52, 53	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ ( -𝐴...𝐴))
55		df-f 6559	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑎):ℕ0⟶( -𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran (coeff‘𝑎) ⊆ ( -𝐴...𝐴)))
56	37, 54, 55	sylanbrc 595	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶( -𝐴...𝐴))
57		fz0ssnn0 13780	. . . . . . . . . . 11 ⊢ (0...𝐴) ⊆ ℕ0
58		fssres 6767	. . . . . . . . . . 11 ⊢ (((coeff‘𝑎):ℕ0⟶( -𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶( -𝐴...𝐴))
59	56, 57, 58	sylancl 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶( -𝐴...𝐴))
60		ovex 7473	. . . . . . . . . . 11 ⊢ ( -𝐴...𝐴) ∈ V
61		ovex 7473	. . . . . . . . . . 11 ⊢ (0...𝐴) ∈ V
62	60, 61	elmap 8924	. . . . . . . . . 10 ⊢ (((coeff‘𝑎) ↾ (0...𝐴)) ∈ (( -𝐴...𝐴) ↑m (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶( -𝐴...𝐴))
63	59, 62	sylibr 237	. . . . . . . . 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 418	. . . . . . . 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 35	. . . . . . 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 1155	. . . . . . . . . 10 ⊢ ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴)
67	66	anim2i 629	. . . . . . . . 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 220	. . . . . . . 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 3021	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝 ↔ 𝑏 ≠ 0𝑝))
70		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏))
71	70	breq1d 5116	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴))
72		fveq2 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏))
73	72	fveq1d 6907	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒))
74	73	fveq2d 6909	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒)))
75	74	breq1d 5116	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
76	75	ralbidv 3189	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
77	69, 71, 76	3anbi123d 1465	. . . . . . . . . 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 3648	. . . . . . . . 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 1155	. . . . . . . . . 10 ⊢ ((𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴)
80	79	anim2i 629	. . . . . . . . 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 220	. . . . . . . 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 787	. . . . . . . . . . . . 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 26543	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (Poly‘ℤ) → 𝑎:ℂ⟶ℂ)
84		ffn 6728	. . . . . . . . . . . . 13 ⊢ (𝑎:ℂ⟶ℂ → 𝑎 Fn ℂ)
85	82, 83, 84	3syl 19	. . . . . . . . . . . 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 789	. . . . . . . . . . . . 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 26543	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (Poly‘ℤ) → 𝑏:ℂ⟶ℂ)
88		ffn 6728	. . . . . . . . . . . . 13 ⊢ (𝑏:ℂ⟶ℂ → 𝑏 Fn ℂ)
89	86, 87, 88	3syl 19	. . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑))
94	93	adantl 487	. . . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑))
96	95	adantl 487	. . . . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . 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 15895	. . . . . . . . . . . . 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 795	. . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . 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 26579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ0)
103		nn0z 12741	. . . . . . . . . . . . . . . . 17 ⊢ ((deg‘𝑎) ∈ ℕ0 → (deg‘𝑎) ∈ ℤ)
104	100, 102, 103	3syl 19	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . 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 12742	. . . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑎) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
108	104, 106, 107	syl2anc 596	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑎) = (deg‘𝑎)
112	33, 111	coeid3 26586	. . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . 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 1257	. . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 26579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (Poly‘ℤ) → (deg‘𝑏) ∈ ℕ0)
119		nn0z 12741	. . . . . . . . . . . . . . . . 17 ⊢ ((deg‘𝑏) ∈ ℕ0 → (deg‘𝑏) ∈ ℤ)
120	115, 118, 119	3syl 19	. . . . . . . . . . . . . . . 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 13003	. . . . . . . . . . . . . . . 16 ⊢ (((deg‘𝑏) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
122	120, 106, 121	syl2anc 596	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (coeff‘𝑏) = (coeff‘𝑏)
125		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (deg‘𝑏) = (deg‘𝑏)
126	124, 125	coeid3 26586	. . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . 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 2808	. . . . . . . . . . . 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 7052	. . . . . . . . . . 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 462	. . . . . . . . . 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 6905	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏))
132	131	reseq1d 5979	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
133	130, 132	impbid1 228	. . . . . . . . 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 419	. . . . . . . 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 619	. . . . . . 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 9045	. . . . . 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 21	. . . . 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 9229	. . . . 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 596	. . . 4 ⊢ (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
140		neeq1 3021	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝 ↔ 𝑐 ≠ 0𝑝))
141		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐))
142	141	breq1d 5116	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴))
143		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐))
144	143	fveq1d 6907	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒))
145	144	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒)))
146	145	breq1d 5116	. . . . . . . . . 10 ⊢ (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
147	146	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
148	140, 142, 147	3anbi123d 1465	. . . . . . . 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 3648	. . . . . . 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 1154	. . . . . . . 8 ⊢ ((𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝)
151	150	anim2i 629	. . . . . . 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 220	. . . . . 6 ⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
153		fveqeq2 6914	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑎 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑎) = 0))
154	153	elrab 3648	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0))
155		plyf 26543	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐:ℂ⟶ℂ)
156	155	ffnd 6729	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Poly‘ℤ) → 𝑐 Fn ℂ)
157	156	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → 𝑐 Fn ℂ)
158		fniniseg 7080	. . . . . . . . . . 11 ⊢ (𝑐 Fn ℂ → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)))
159	157, 158	syl 18	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)))
160	154, 159	bitr4id 293	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ 𝑎 ∈ (◡𝑐 “ {0})))
161	160	eqrdv 2761	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = (◡𝑐 “ {0}))
162		eqid 2763	. . . . . . . . . 10 ⊢ (◡𝑐 “ {0}) = (◡𝑐 “ {0})
163	162	fta1 26656	. . . . . . . . 9 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → ((◡𝑐 “ {0}) ∈ Fin ∧ (♯‘(◡𝑐 “ {0})) ≤ (deg‘𝑐)))
164	163	simpld 500	. . . . . . . 8 ⊢ ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (◡𝑐 “ {0}) ∈ Fin)
165	161, 164	eqeltrd 2864	. . . . . . 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 35	. . . . 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 3157	. . . 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 9357	. . . 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 596	. . 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 2869	. 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 2864	1 ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  {csn 4588  ∪ ciun 4954   class class class wbr 5106   ↦ cmpt 5189  ^◡ccnv 5654  ran crn 5656   ↾ cres 5657   “ cima 5658   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872   ≼ cdom 8996  Fincfn 8998  ℂcc 11222  ℝcr 11223  0cc0 11224   · cmul 11229   ≤ cle 11368   -cneg 11566  ℕ₀cn0 12630  ℤcz 12717  ℤ_≥cuz 12989  ...cfz 13663  ↑cexp 14228  ♯chash 14498  abscabs 15427  Σcsu 15879  0_𝑝c0p 26017  Polycply 26529  coeffccoe 26531  degcdgr 26532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-oadd 8505  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-inf 9459  df-oi 9528  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-rp 13145  df-fz 13664  df-fzo 13813  df-fl 13956  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-rlim 15682  df-sum 15880  df-0p 26018  df-ply 26533  df-idp 26534  df-coe 26535  df-dgr 26536  df-quot 26639

This theorem is used by:  aannenlem3  26684