Theorem aannenlem2 25834

Description: Lemma for aannen 25836. (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
aannenlem2	⊢ 𝔸 = ∪ ran 𝐻

Distinct variable group:   𝑎,𝑏,𝑐,𝑑,𝑒

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

Proof of Theorem aannenlem2

Dummy variables 𝑓 𝑔 ℎ 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveqeq2 6898	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑔 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑔) = 0))
2	1	rexbidv 3179	. . . . . . . . . 10 ⊢ (𝑏 = 𝑔 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑔) = 0))
3		simp3 1139	. . . . . . . . . 10 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ)
4		neeq1 3004	. . . . . . . . . . . . 13 ⊢ (𝑑 = ℎ → (𝑑 ≠ 0𝑝 ↔ ℎ ≠ 0𝑝))
5		fveq2 6889	. . . . . . . . . . . . . 14 ⊢ (𝑑 = ℎ → (deg‘𝑑) = (deg‘ℎ))
6	5	breq1d 5158	. . . . . . . . . . . . 13 ⊢ (𝑑 = ℎ → ((deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ↔ (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
7		fveq2 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ℎ → (coeff‘𝑑) = (coeff‘ℎ))
8	7	fveq1d 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → ((coeff‘𝑑)‘𝑒) = ((coeff‘ℎ)‘𝑒))
9	8	fveq2d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ℎ → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒)))
10	9	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑑 = ℎ → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ↔ (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
11	10	ralbidv 3178	. . . . . . . . . . . . 13 ⊢ (𝑑 = ℎ → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
12	4, 6, 11	3anbi123d 1437	. . . . . . . . . . . 12 ⊢ (𝑑 = ℎ → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )) ↔ (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))))
13		eldifi 4126	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ℎ ∈ (Poly‘ℤ))
14	13	adantr 482	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ))
15	14	3adant2 1132	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ))
16		eldifsni 4793	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ℎ ≠ 0𝑝)
17	16	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ≠ 0𝑝)
18		0nn0 12484	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
19		dgrcl 25739	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ∈ (Poly‘ℤ) → (deg‘ℎ) ∈ ℕ0)
20	14, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘ℎ) ∈ ℕ0)
21		prssi 4824	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ (deg‘ℎ) ∈ ℕ0) → {0, (deg‘ℎ)} ⊆ ℕ0)
22	18, 20, 21	sylancr 588	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘ℎ)} ⊆ ℕ0)
23		ssrab2 4077	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ⊆ ℕ0
24	23	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ⊆ ℕ0)
25	22, 24	unssd 4186	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℕ0)
26		nn0ssre 12473	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ⊆ ℝ
27		ressxr 11255	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℝ*
28	26, 27	sstri 3991	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ⊆ ℝ*
29	25, 28	sstrdi 3994	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*)
30		fvex 6902	. . . . . . . . . . . . . . . . 17 ⊢ (deg‘ℎ) ∈ V
31	30	prid2 4767	. . . . . . . . . . . . . . . 16 ⊢ (deg‘ℎ) ∈ {0, (deg‘ℎ)}
32		elun1 4176	. . . . . . . . . . . . . . . 16 ⊢ ((deg‘ℎ) ∈ {0, (deg‘ℎ)} → (deg‘ℎ) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
33	31, 32	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (deg‘ℎ) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))})
34		supxrub 13300	. . . . . . . . . . . . . . 15 ⊢ ((({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ* ∧ (deg‘ℎ) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))})) → (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))
35	29, 33, 34	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))
36	29	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*)
37		fveq2 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘0))
38		abs0 15229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs‘0) = 0
39	37, 38	eqtrdi 2789	. . . . . . . . . . . . . . . . . . 19 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = 0)
40		c0ex 11205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
41	40	prid1 4766	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ {0, (deg‘ℎ)}
42		elun1 4176	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ {0, (deg‘ℎ)} → 0 ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))})
44	39, 43	eqeltrdi 2842	. . . . . . . . . . . . . . . . . 18 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
45	44	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) = 0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
46		eqeq1 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (abs‘((coeff‘ℎ)‘𝑒)) → (𝑔 = (abs‘((coeff‘ℎ)‘𝑖)) ↔ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖))))
47	46	rexbidv 3179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = (abs‘((coeff‘ℎ)‘𝑒)) → (∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖)) ↔ ∃𝑖 ∈ (0...(deg‘ℎ))(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖))))
48		0z 12566	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℤ
49		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (coeff‘ℎ) = (coeff‘ℎ)
50	49	coef2 25737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘ℎ):ℕ0⟶ℤ)
51	14, 48, 50	sylancl 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘ℎ):ℕ0⟶ℤ)
52	51	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘ℎ)‘𝑒) ∈ ℤ)
53		nn0abscl 15256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((coeff‘ℎ)‘𝑒) ∈ ℤ → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ℕ0)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ℕ0)
55	54	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ℕ0)
56		simplr 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ∈ ℕ0)
57	20	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (deg‘ℎ) ∈ ℕ0)
58	14	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ℎ ∈ (Poly‘ℤ))
59		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ((coeff‘ℎ)‘𝑒) ≠ 0)
60		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (deg‘ℎ) = (deg‘ℎ)
61	49, 60	dgrub 25740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ 𝑒 ∈ ℕ0 ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ))
62	58, 56, 59, 61	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ))
63		elfz2nn0 13589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 ∈ (0...(deg‘ℎ)) ↔ (𝑒 ∈ ℕ0 ∧ (deg‘ℎ) ∈ ℕ0 ∧ 𝑒 ≤ (deg‘ℎ)))
64	56, 57, 62, 63	syl3anbrc 1344	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ∈ (0...(deg‘ℎ)))
65		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒))
66		2fveq3 6894	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑒 → (abs‘((coeff‘ℎ)‘𝑖)) = (abs‘((coeff‘ℎ)‘𝑒)))
67	66	rspceeqv 3633	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑒 ∈ (0...(deg‘ℎ)) ∧ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒))) → ∃𝑖 ∈ (0...(deg‘ℎ))(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖)))
68	64, 65, 67	sylancl 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ∃𝑖 ∈ (0...(deg‘ℎ))(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖)))
69	47, 55, 68	elrabd 3685	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))})
70		elun2 4177	. . . . . . . . . . . . . . . . . 18 ⊢ ((abs‘((coeff‘ℎ)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
71	69, 70	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
72	45, 71	pm2.61dane 3030	. . . . . . . . . . . . . . . 16 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
73		supxrub 13300	. . . . . . . . . . . . . . . 16 ⊢ ((({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ* ∧ (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))})) → (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))
74	36, 72, 73	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))
75	74	ralrimiva 3147	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))
76	17, 35, 75	3jca 1129	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
77	76	3adant2 1132	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
78	12, 15, 77	elrabd 3685	. . . . . . . . . . 11 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))})
79		simp2 1138	. . . . . . . . . . 11 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (ℎ‘𝑔) = 0)
80		fveq1 6888	. . . . . . . . . . . . 13 ⊢ (𝑐 = ℎ → (𝑐‘𝑔) = (ℎ‘𝑔))
81	80	eqeq1d 2735	. . . . . . . . . . . 12 ⊢ (𝑐 = ℎ → ((𝑐‘𝑔) = 0 ↔ (ℎ‘𝑔) = 0))
82	81	rspcev 3613	. . . . . . . . . . 11 ⊢ ((ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} ∧ (ℎ‘𝑔) = 0) → ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑔) = 0)
83	78, 79, 82	syl2anc 585	. . . . . . . . . 10 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑔) = 0)
84	2, 3, 83	elrabd 3685	. . . . . . . . 9 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0})
85		prfi 9319	. . . . . . . . . . . . . 14 ⊢ {0, (deg‘ℎ)} ∈ Fin
86		fzfi 13934	. . . . . . . . . . . . . . . 16 ⊢ (0...(deg‘ℎ)) ∈ Fin
87		abrexfi 9349	. . . . . . . . . . . . . . . 16 ⊢ ((0...(deg‘ℎ)) ∈ Fin → {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin)
88	86, 87	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin
89		rabssab 4083	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}
90		ssfi 9170	. . . . . . . . . . . . . . 15 ⊢ (({𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin ∧ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) → {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin)
91	88, 89, 90	mp2an 691	. . . . . . . . . . . . . 14 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin
92		unfi 9169	. . . . . . . . . . . . . 14 ⊢ (({0, (deg‘ℎ)} ∈ Fin ∧ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin)
93	85, 91, 92	mp2an 691	. . . . . . . . . . . . 13 ⊢ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin
94	33	ne0ii 4337	. . . . . . . . . . . . 13 ⊢ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅
95		xrltso 13117	. . . . . . . . . . . . . 14 ⊢ < Or ℝ*
96		fisupcl 9461	. . . . . . . . . . . . . 14 ⊢ (( < Or ℝ* ∧ (({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅ ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*)) → sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
97	95, 96	mpan 689	. . . . . . . . . . . . 13 ⊢ ((({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅ ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*) → sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
98	93, 94, 29, 97	mp3an12i 1466	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}))
99	25, 98	sseldd 3983	. . . . . . . . . . 11 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
100	99	3adant2 1132	. . . . . . . . . 10 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ℕ0)
101		eqidd 2734	. . . . . . . . . 10 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0})
102		breq2 5152	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
103		breq2 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
104	103	ralbidv 3178	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )))
105	102, 104	3anbi23d 1440	. . . . . . . . . . . . . 14 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))))
106	105	rabbidv 3441	. . . . . . . . . . . . 13 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} = {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))})
107	106	rexeqdv 3327	. . . . . . . . . . . 12 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → (∃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) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0))
108	107	rabbidv 3441	. . . . . . . . . . 11 ⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) → {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) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0})
109	108	rspceeqv 3633	. . . . . . . . . 10 ⊢ ((sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ℕ0 ∧ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0}) → ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})
110	100, 101, 109	syl2anc 585	. . . . . . . . 9 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})
111		cnex 11188	. . . . . . . . . . 11 ⊢ ℂ ∈ V
112	111	rabex 5332	. . . . . . . . . 10 ⊢ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} ∈ V
113		eleq2 2823	. . . . . . . . . . 11 ⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 ↔ 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0}))
114		eqeq1 2737	. . . . . . . . . . . 12 ⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → (𝑓 = {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) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
115	114	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → (∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} ↔ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
116	113, 115	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → ((𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) ↔ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} ∧ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})))
117	112, 116	spcev 3597	. . . . . . . . 9 ⊢ ((𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} ∧ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
118	84, 110, 117	syl2anc 585	. . . . . . . 8 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
119	118	3exp 1120	. . . . . . 7 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ((ℎ‘𝑔) = 0 → (𝑔 ∈ ℂ → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))))
120	119	rexlimiv 3149	. . . . . 6 ⊢ (∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0 → (𝑔 ∈ ℂ → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})))
121	120	impcom 409	. . . . 5 ⊢ ((𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0) → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
122		eleq2 2823	. . . . . . . . 9 ⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 ↔ 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
123	1	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑔 → (∃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‘𝑔) = 0))
124	123	elrab 3683	. . . . . . . . . 10 ⊢ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} ↔ (𝑔 ∈ ℂ ∧ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0))
125		simp1 1137	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎) → ℎ ≠ 0𝑝)
126	125	anim2i 618	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)) → (ℎ ∈ (Poly‘ℤ) ∧ ℎ ≠ 0𝑝))
127	5	breq1d 5158	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘ℎ) ≤ 𝑎))
128	9	breq1d 5158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ℎ → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))
129	128	ralbidv 3178	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))
130	4, 127, 129	3anbi123d 1437	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ℎ → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)))
131	130	elrab 3683	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ↔ (ℎ ∈ (Poly‘ℤ) ∧ (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)))
132		eldifsn 4790	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ (ℎ ∈ (Poly‘ℤ) ∧ ℎ ≠ 0𝑝))
133	126, 131, 132	3imtr4i 292	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}))
134	133	ssriv 3986	. . . . . . . . . . . 12 ⊢ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
135		ssrexv 4051	. . . . . . . . . . . . 13 ⊢ ({𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝}) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0 → ∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐‘𝑔) = 0))
136	81	cbvrexvw 3236	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐‘𝑔) = 0 ↔ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)
137	135, 136	imbitrdi 250	. . . . . . . . . . . 12 ⊢ ({𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝}) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0 → ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
138	134, 137	ax-mp 5	. . . . . . . . . . 11 ⊢ (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0 → ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)
139	138	anim2i 618	. . . . . . . . . 10 ⊢ ((𝑔 ∈ ℂ ∧ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0) → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
140	124, 139	sylbi 216	. . . . . . . . 9 ⊢ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
141	122, 140	syl6bi 253	. . . . . . . 8 ⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)))
142	141	rexlimivw 3152	. . . . . . 7 ⊢ (∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)))
143	142	impcom 409	. . . . . 6 ⊢ ((𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
144	143	exlimiv 1934	. . . . 5 ⊢ (∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
145	121, 144	impbii 208	. . . 4 ⊢ ((𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0) ↔ ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
146		elaa 25821	. . . 4 ⊢ (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
147		eluniab 4923	. . . 4 ⊢ (𝑔 ∈ ∪ {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}} ↔ ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))
148	145, 146, 147	3bitr4i 303	. . 3 ⊢ (𝑔 ∈ 𝔸 ↔ 𝑔 ∈ ∪ {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}})
149	148	eqriv 2730	. 2 ⊢ 𝔸 = ∪ {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}}
150		aannenlem.a	. . . 4 ⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})
151	150	rnmpt 5953	. . 3 ⊢ ran 𝐻 = {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}}
152	151	unieqi 4921	. 2 ⊢ ∪ ran 𝐻 = ∪ {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}}
153	149, 152	eqtr4i 2764	1 ⊢ 𝔸 = ∪ ran 𝐻

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433   ∖ cdif 3945   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  {csn 4628  {cpr 4630  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231   Or wor 5587  ran crn 5677  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406  Fincfn 8936  supcsup 9432  ℂcc 11105  ℝcr 11106  0cc0 11107  ℝ^*cxr 11244   < clt 11245   ≤ cle 11246  ℕ₀cn0 12469  ℤcz 12555  ...cfz 13481  abscabs 15178  0_𝑝c0p 25178  Polycply 25690  coeffccoe 25692  degcdgr 25693  𝔸caa 25819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-sum 15630  df-0p 25179  df-ply 25694  df-coe 25696  df-dgr 25697  df-aa 25820

This theorem is referenced by:  aannenlem3  25835