Theorem aannenlem2 24921

Description: Lemma for aannen 24923. (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 6682	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑔 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑔) = 0))
2	1	rexbidv 3300	. . . . . . . . . 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 1134	. . . . . . . . . 10 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ)
4		neeq1 3081	. . . . . . . . . . . . 13 ⊢ (𝑑 = ℎ → (𝑑 ≠ 0𝑝 ↔ ℎ ≠ 0𝑝))
5		fveq2 6673	. . . . . . . . . . . . . 14 ⊢ (𝑑 = ℎ → (deg‘𝑑) = (deg‘ℎ))
6	5	breq1d 5079	. . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ℎ → (coeff‘𝑑) = (coeff‘ℎ))
8	7	fveq1d 6675	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → ((coeff‘𝑑)‘𝑒) = ((coeff‘ℎ)‘𝑒))
9	8	fveq2d 6677	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ℎ → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒)))
10	9	breq1d 5079	. . . . . . . . . . . . . 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 3200	. . . . . . . . . . . . 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 1432	. . . . . . . . . . . 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 4106	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ℎ ∈ (Poly‘ℤ))
14	13	adantr 483	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ))
15	14	3adant2 1127	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ))
16		eldifsni 4725	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ℎ ≠ 0𝑝)
17	16	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ≠ 0𝑝)
18		0nn0 11915	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
19		dgrcl 24826	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ∈ (Poly‘ℤ) → (deg‘ℎ) ∈ ℕ0)
20	14, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘ℎ) ∈ ℕ0)
21		prssi 4757	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ (deg‘ℎ) ∈ ℕ0) → {0, (deg‘ℎ)} ⊆ ℕ0)
22	18, 20, 21	sylancr 589	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘ℎ)} ⊆ ℕ0)
23		ssrab2 4059	. . . . . . . . . . . . . . . . . 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 4165	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℕ0)
26		nn0ssre 11904	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ⊆ ℝ
27		ressxr 10688	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℝ*
28	26, 27	sstri 3979	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ⊆ ℝ*
29	25, 28	sstrdi 3982	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*)
30		fvex 6686	. . . . . . . . . . . . . . . . 17 ⊢ (deg‘ℎ) ∈ V
31	30	prid2 4702	. . . . . . . . . . . . . . . 16 ⊢ (deg‘ℎ) ∈ {0, (deg‘ℎ)}
32		elun1 4155	. . . . . . . . . . . . . . . 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 12720	. . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘0))
38		abs0 14648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs‘0) = 0
39	37, 38	syl6eq 2875	. . . . . . . . . . . . . . . . . . 19 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = 0)
40		c0ex 10638	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
41	40	prid1 4701	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ {0, (deg‘ℎ)}
42		elun1 4155	. . . . . . . . . . . . . . . . . . . 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 2924	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (abs‘((coeff‘ℎ)‘𝑒)) → (𝑔 = (abs‘((coeff‘ℎ)‘𝑖)) ↔ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖))))
47	46	rexbidv 3300	. . . . . . . . . . . . . . . . . . 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 11995	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℤ
49		eqid 2824	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (coeff‘ℎ) = (coeff‘ℎ)
50	49	coef2 24824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘ℎ):ℕ0⟶ℤ)
51	14, 48, 50	sylancl 588	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘ℎ):ℕ0⟶ℤ)
52	51	ffvelrnda 6854	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘ℎ)‘𝑒) ∈ ℤ)
53		nn0abscl 14675	. . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ℕ0)
56		simplr 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ∈ ℕ0)
57	20	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (deg‘ℎ) ∈ ℕ0)
58	14	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ℎ ∈ (Poly‘ℤ))
59		simpr 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ((coeff‘ℎ)‘𝑒) ≠ 0)
60		eqid 2824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (deg‘ℎ) = (deg‘ℎ)
61	49, 60	dgrub 24827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ 𝑒 ∈ ℕ0 ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ))
62	58, 56, 59, 61	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ))
63		elfz2nn0 13001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 ∈ (0...(deg‘ℎ)) ↔ (𝑒 ∈ ℕ0 ∧ (deg‘ℎ) ∈ ℕ0 ∧ 𝑒 ≤ (deg‘ℎ)))
64	56, 57, 62, 63	syl3anbrc 1339	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ∈ (0...(deg‘ℎ)))
65		eqid 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒))
66		2fveq3 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑒 → (abs‘((coeff‘ℎ)‘𝑖)) = (abs‘((coeff‘ℎ)‘𝑒)))
67	66	rspceeqv 3641	. . . . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . . . . 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 4156	. . . . . . . . . . . . . . . . . 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 3107	. . . . . . . . . . . . . . . 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 12720	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . 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 1124	. . . . . . . . . . . . 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 1127	. . . . . . . . . . . 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 1133	. . . . . . . . . . 11 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (ℎ‘𝑔) = 0)
80		fveq1 6672	. . . . . . . . . . . . 13 ⊢ (𝑐 = ℎ → (𝑐‘𝑔) = (ℎ‘𝑔))
81	80	eqeq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑐 = ℎ → ((𝑐‘𝑔) = 0 ↔ (ℎ‘𝑔) = 0))
82	81	rspcev 3626	. . . . . . . . . . 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 586	. . . . . . . . . 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 8796	. . . . . . . . . . . . . 14 ⊢ {0, (deg‘ℎ)} ∈ Fin
86		fzfi 13343	. . . . . . . . . . . . . . . 16 ⊢ (0...(deg‘ℎ)) ∈ Fin
87		abrexfi 8827	. . . . . . . . . . . . . . . 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 4063	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}
90		ssfi 8741	. . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . 14 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin
92		unfi 8788	. . . . . . . . . . . . . 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 690	. . . . . . . . . . . . 13 ⊢ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin
94	33	ne0ii 4306	. . . . . . . . . . . . 13 ⊢ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅
95		xrltso 12537	. . . . . . . . . . . . . 14 ⊢ < Or ℝ*
96		fisupcl 8936	. . . . . . . . . . . . . 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 688	. . . . . . . . . . . . 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 1461	. . . . . . . . . . . 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 3971	. . . . . . . . . . 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 1127	. . . . . . . . . 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 2825	. . . . . . . . . 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 5073	. . . . . . . . . . . . . . 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 5073	. . . . . . . . . . . . . . . 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 3200	. . . . . . . . . . . . . . 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 1435	. . . . . . . . . . . . . 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 3483	. . . . . . . . . . . . 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 3419	. . . . . . . . . . . 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 3483	. . . . . . . . . . 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 3641	. . . . . . . . . 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 586	. . . . . . . . 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 10621	. . . . . . . . . . 11 ⊢ ℂ ∈ V
112	111	rabex 5238	. . . . . . . . . 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 2904	. . . . . . . . . . 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 2828	. . . . . . . . . . . 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 3300	. . . . . . . . . . 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 3610	. . . . . . . . 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 586	. . . . . . . 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 1115	. . . . . . 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 3283	. . . . . 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 410	. . . . 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 2904	. . . . . . . . 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 3300	. . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 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 5079	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘ℎ) ≤ 𝑎))
128	9	breq1d 5079	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ℎ → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))
129	128	ralbidv 3200	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))
130	4, 127, 129	3anbi123d 1432	. . . . . . . . . . . . . . 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 4722	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ (ℎ ∈ (Poly‘ℤ) ∧ ℎ ≠ 0𝑝))
133	126, 131, 132	3imtr4i 294	. . . . . . . . . . . . 13 ⊢ (ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}))
134	133	ssriv 3974	. . . . . . . . . . . 12 ⊢ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
135		ssrexv 4037	. . . . . . . . . . . . 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 3453	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐‘𝑔) = 0 ↔ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)
137	135, 136	syl6ib 253	. . . . . . . . . . . 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 219	. . . . . . . . 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 255	. . . . . . . 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 3285	. . . . . . 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 410	. . . . . 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 1930	. . . . 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 211	. . . 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 24908	. . . 4 ⊢ (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
147		eluniab 4856	. . . 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 305	. . 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 2821	. 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 5830	. . 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 4854	. 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 2850	1 ⊢ 𝔸 = ∪ ran 𝐻

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  {crab 3145   ∖ cdif 3936   ∪ cun 3937   ⊆ wss 3939  ∅c0 4294  {csn 4570  {cpr 4572  ∪ cuni 4841   class class class wbr 5069   ↦ cmpt 5149   Or wor 5476  ran crn 5559  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  Fincfn 8512  supcsup 8907  ℂcc 10538  ℝcr 10539  0cc0 10540  ℝ^*cxr 10677   < clt 10678   ≤ cle 10679  ℕ₀cn0 11900  ℤcz 11984  ...cfz 12895  abscabs 14596  0_𝑝c0p 24273  Polycply 24777  coeffccoe 24779  degcdgr 24780  𝔸caa 24906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-sup 8909  df-inf 8910  df-oi 8977  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-clim 14848  df-rlim 14849  df-sum 15046  df-0p 24274  df-ply 24781  df-coe 24783  df-dgr 24784  df-aa 24907

This theorem is referenced by:  aannenlem3  24922