Theorem dgrlem 26204

Description: Lemma for dgrcl 26208 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)

Hypothesis

Ref	Expression
dgrval.1	⊢ 𝐴 = (coeff‘𝐹)

Assertion

Ref	Expression
dgrlem	⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))

Distinct variable groups:   𝑥,𝑛,𝐴   𝑛,𝐹,𝑥   𝑆,𝑛,𝑥

Proof of Theorem dgrlem

Dummy variables 𝑎 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply2 26171	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
2	1	simprbi 497	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
3		simplrr 778	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))
4		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝐹 ∈ (Poly‘𝑆))
5		plybss 26169	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝑆 ⊆ ℂ)
7		0cnd 11128	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 0 ∈ ℂ)
8	7	snssd 4753	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → {0} ⊆ ℂ)
9	6, 8	unssd 4133	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑆 ∪ {0}) ⊆ ℂ)
10		cnex 11110	. . . . . . . . 9 ⊢ ℂ ∈ V
11		ssexg 5260	. . . . . . . . 9 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
12	9, 10, 11	sylancl 587	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑆 ∪ {0}) ∈ V)
13		nn0ex 12434	. . . . . . . 8 ⊢ ℕ0 ∈ V
14		elmapg 8779	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
15	12, 13, 14	sylancl 587	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
16	3, 15	mpbid 232	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
17		dgrval.1	. . . . . . . 8 ⊢ 𝐴 = (coeff‘𝐹)
18		simplrl 777	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝑛 ∈ ℕ0)
19	16, 9	fssd 6679	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝑎:ℕ0⟶ℂ)
20		simprl 771	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0})
21		simprr 773	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
22	4, 18, 19, 20, 21	coeeq 26202	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (coeff‘𝐹) = 𝑎)
23	17, 22	eqtr2id 2785	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝑎 = 𝐴)
24	23	feq1d 6644	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0})))
25	16, 24	mpbid 232	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
26	25	ex 412	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → 𝐴:ℕ0⟶(𝑆 ∪ {0})))
27	26	rexlimdvva 3195	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → 𝐴:ℕ0⟶(𝑆 ∪ {0})))
28	2, 27	mpd 15	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
29		nn0ssz 12538	. . 3 ⊢ ℕ0 ⊆ ℤ
30		ffn 6662	. . . . . . . . . . . . . 14 ⊢ (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31		elpreima 7004	. . . . . . . . . . . . . 14 ⊢ (𝑎 Fn ℕ0 → (𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℕ0 ∧ (𝑎‘𝑥) ∈ (ℂ ∖ {0}))))
32	19, 30, 31	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℕ0 ∧ (𝑎‘𝑥) ∈ (ℂ ∖ {0}))))
33	32	biimpa 476	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0}))) → (𝑥 ∈ ℕ0 ∧ (𝑎‘𝑥) ∈ (ℂ ∖ {0})))
34		eldifsni 4734	. . . . . . . . . . . 12 ⊢ ((𝑎‘𝑥) ∈ (ℂ ∖ {0}) → (𝑎‘𝑥) ≠ 0)
35	33, 34	simpl2im 503	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0}))) → (𝑎‘𝑥) ≠ 0)
36	33	simpld 494	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0}))) → 𝑥 ∈ ℕ0)
37		plyco0 26167	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛)))
38	18, 19, 37	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛)))
39	20, 38	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → ∀𝑥 ∈ ℕ0 ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛))
40	39	r19.21bi 3230	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑥 ∈ ℕ0) → ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛))
41	36, 40	syldan 592	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0}))) → ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛))
42	35, 41	mpd 15	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0}))) → 𝑥 ≤ 𝑛)
43	42	ralrimiva 3130	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → ∀𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
44	23	cnveqd 5824	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → ◡𝑎 = ◡𝐴)
45	44	imaeq1d 6018	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (◡𝑎 “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0})))
46	43, 45	raleqtrdv 3298	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) ∧ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
47	46	ex 412	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
48	47	expr 456	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)))
49	48	rexlimdv 3137	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
50	49	reximdva 3151	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → ∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
51	2, 50	mpd 15	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
52		ssrexv 3992	. . 3 ⊢ (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
53	29, 51, 52	mpsyl 68	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
54	28, 53	jca 511	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  {csn 4568   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5623   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  ℂcc 11027  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034   ≤ cle 11171  ℕ₀cn0 12428  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  ↑cexp 14014  Σcsu 15639  Polycply 26159  coeffccoe 26161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-rlim 15442  df-sum 15640  df-0p 25647  df-ply 26163  df-coe 26165

This theorem is referenced by:  coef  26205  dgrcl  26208  dgrub  26209  dgrlb  26211