Theorem aannenlem2 26291

Description: Lemma for aannen 26293. (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 6841	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑔 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑔) = 0))
2	1	rexbidv 3158	. . . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ)
4		neeq1 2992	. . . . . . . . . . . . 13 ⊢ (𝑑 = ℎ → (𝑑 ≠ 0𝑝 ↔ ℎ ≠ 0𝑝))
5		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑑 = ℎ → (deg‘𝑑) = (deg‘ℎ))
6	5	breq1d 5106	. . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ℎ → (coeff‘𝑑) = (coeff‘ℎ))
8	7	fveq1d 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → ((coeff‘𝑑)‘𝑒) = ((coeff‘ℎ)‘𝑒))
9	8	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ℎ → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒)))
10	9	breq1d 5106	. . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . 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 1438	. . . . . . . . . . . 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 4081	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ℎ ∈ (Poly‘ℤ))
14	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ))
15	14	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ))
16		eldifsni 4744	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) → ℎ ≠ 0𝑝)
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ≠ 0𝑝)
18		0nn0 12414	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℕ0
19		dgrcl 26192	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ ∈ (Poly‘ℤ) → (deg‘ℎ) ∈ ℕ0)
20	14, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘ℎ) ∈ ℕ0)
21		prssi 4775	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ (deg‘ℎ) ∈ ℕ0) → {0, (deg‘ℎ)} ⊆ ℕ0)
22	18, 20, 21	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘ℎ)} ⊆ ℕ0)
23		ssrab2 4030	. . . . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℕ0)
26		nn0ssre 12403	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ⊆ ℝ
27		ressxr 11174	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℝ*
28	26, 27	sstri 3941	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ⊆ ℝ*
29	25, 28	sstrdi 3944	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*)
30		fvex 6845	. . . . . . . . . . . . . . . . 17 ⊢ (deg‘ℎ) ∈ V
31	30	prid2 4718	. . . . . . . . . . . . . . . 16 ⊢ (deg‘ℎ) ∈ {0, (deg‘ℎ)}
32		elun1 4132	. . . . . . . . . . . . . . . 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 13237	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘0))
38		abs0 15206	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs‘0) = 0
39	37, 38	eqtrdi 2785	. . . . . . . . . . . . . . . . . . 19 ⊢ (((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = 0)
40		c0ex 11124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
41	40	prid1 4717	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ {0, (deg‘ℎ)}
42		elun1 4132	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (abs‘((coeff‘ℎ)‘𝑒)) → (𝑔 = (abs‘((coeff‘ℎ)‘𝑖)) ↔ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖))))
47	46	rexbidv 3158	. . . . . . . . . . . . . . . . . . 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 12497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℤ
49		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (coeff‘ℎ) = (coeff‘ℎ)
50	49	coef2 26190	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘ℎ):ℕ0⟶ℤ)
51	14, 48, 50	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘ℎ):ℕ0⟶ℤ)
52	51	ffvelcdmda 7027	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ((coeff‘ℎ)‘𝑒) ∈ ℤ)
53		nn0abscl 15233	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → (deg‘ℎ) ∈ ℕ0)
58	14	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ℎ ∈ (Poly‘ℤ))
59		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → ((coeff‘ℎ)‘𝑒) ≠ 0)
60		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (deg‘ℎ) = (deg‘ℎ)
61	49, 60	dgrub 26193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ 𝑒 ∈ ℕ0 ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ))
62	58, 56, 59, 61	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ))
63		elfz2nn0 13532	. . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒))
66		2fveq3 6837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑒 → (abs‘((coeff‘ℎ)‘𝑖)) = (abs‘((coeff‘ℎ)‘𝑒)))
67	66	rspceeqv 3597	. . . . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . . . 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 3646	. . . . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . . . . 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 13237	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 3126	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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 3646	. . . . . . . . . . 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 1137	. . . . . . . . . . 11 ⊢ ((ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (ℎ‘𝑔) = 0)
80		fveq1 6831	. . . . . . . . . . . . 13 ⊢ (𝑐 = ℎ → (𝑐‘𝑔) = (ℎ‘𝑔))
81	80	eqeq1d 2736	. . . . . . . . . . . 12 ⊢ (𝑐 = ℎ → ((𝑐‘𝑔) = 0 ↔ (ℎ‘𝑔) = 0))
82	81	rspcev 3574	. . . . . . . . . . 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 584	. . . . . . . . . 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 3646	. . . . . . . . 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 9222	. . . . . . . . . . . . . 14 ⊢ {0, (deg‘ℎ)} ∈ Fin
86		fzfi 13893	. . . . . . . . . . . . . . . 16 ⊢ (0...(deg‘ℎ)) ∈ Fin
87		abrexfi 9250	. . . . . . . . . . . . . . . 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 4035	. . . . . . . . . . . . . . 15 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}
90		ssfi 9095	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . 14 ⊢ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin
92		unfi 9093	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 13 ⊢ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin
94	33	ne0ii 4294	. . . . . . . . . . . . 13 ⊢ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅
95		xrltso 13053	. . . . . . . . . . . . . 14 ⊢ < Or ℝ*
96		fisupcl 9371	. . . . . . . . . . . . . 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 690	. . . . . . . . . . . . 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 1467	. . . . . . . . . . . 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 3932	. . . . . . . . . . 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 1131	. . . . . . . . . 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 2735	. . . . . . . . . 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 5100	. . . . . . . . . . . . . . 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 5100	. . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . 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 1441	. . . . . . . . . . . . . 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 3404	. . . . . . . . . . . . 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 3295	. . . . . . . . . . . 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 3404	. . . . . . . . . . 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 3597	. . . . . . . . . 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 584	. . . . . . . . 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 11105	. . . . . . . . . . 11 ⊢ ℂ ∈ V
112	111	rabex 5282	. . . . . . . . . 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 2738	. . . . . . . . . . . 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 3158	. . . . . . . . . . 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 3558	. . . . . . . . 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 584	. . . . . . . 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 1119	. . . . . . 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 3128	. . . . . 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 407	. . . . 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 3158	. . . . . . . . . . 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 3644	. . . . . . . . . 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 1136	. . . . . . . . . . . . . . 15 ⊢ ((ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎) → ℎ ≠ 0𝑝)
126	125	anim2i 617	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ (Poly‘ℤ) ∧ (ℎ ≠ 0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)) → (ℎ ∈ (Poly‘ℤ) ∧ ℎ ≠ 0𝑝))
127	5	breq1d 5106	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘ℎ) ≤ 𝑎))
128	9	breq1d 5106	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = ℎ → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))
129	128	ralbidv 3157	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ℎ → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))
130	4, 127, 129	3anbi123d 1438	. . . . . . . . . . . . . . 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 3644	. . . . . . . . . . . . . 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 4740	. . . . . . . . . . . . . 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 3935	. . . . . . . . . . . 12 ⊢ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖ {0𝑝})
135		ssrexv 4001	. . . . . . . . . . . . 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 3213	. . . . . . . . . . . . 13 ⊢ (∃𝑐 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑐‘𝑔) = 0 ↔ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)
137	135, 136	imbitrdi 251	. . . . . . . . . . . 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 617	. . . . . . . . . 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 217	. . . . . . . . 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	biimtrdi 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 3131	. . . . . . 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 407	. . . . . 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 1931	. . . . 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 209	. . . 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 26278	. . . 4 ⊢ (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0))
147		eluniab 4875	. . . 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 2731	. 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 5904	. . 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 4873	. 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 2760	1 ⊢ 𝔸 = ∪ ran 𝐻

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2712   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  {crab 3397   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4283  {csn 4578  {cpr 4580  ∪ cuni 4861   class class class wbr 5096   ↦ cmpt 5177   Or wor 5529  ran crn 5623  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  Fincfn 8881  supcsup 9341  ℂcc 11022  ℝcr 11023  0cc0 11024  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165  ℕ₀cn0 12399  ℤcz 12486  ...cfz 13421  abscabs 15155  0_𝑝c0p 25624  Polycply 26143  coeffccoe 26145  degcdgr 26146  𝔸caa 26276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-fzo 13569  df-fl 13710  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-rlim 15410  df-sum 15608  df-0p 25625  df-ply 26147  df-coe 26149  df-dgr 26150  df-aa 26277

This theorem is referenced by:  aannenlem3  26292