Theorem dgrlem 26528

Description: Lemma for dgrcl 26532 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 26494	. . . 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 503	. . 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 790	. . . . . . 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 779	. . . . . . . . . . 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 26492	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
6	4, 5	syl 18	. . . . . . . . . 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 11280	. . . . . . . . . . 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 4747	. . . . . . . . . 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 4138	. . . . . . . . 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 11262	. . . . . . . . 9 ⊢ ℂ ∈ V
11		ssexg 5281	. . . . . . . . 9 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
12	9, 10, 11	sylancl 598	. . . . . . . 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 12593	. . . . . . . 8 ⊢ ℕ0 ∈ V
14		elmapg 8843	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
15	12, 13, 14	sylancl 598	. . . . . . 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 235	. . . . . 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 789	. . . . . . . . 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 6719	. . . . . . . . 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 783	. . . . . . . . 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 785	. . . . . . . . 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 26526	. . . . . . . 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 2809	. . . . . . 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 6683	. . . . . 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 235	. . . . 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 418	. . . 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 3220	. . 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 16	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
29		nn0ssz 12697	. . 3 ⊢ ℕ0 ⊆ ℤ
30		ffn 6701	. . . . . . . . . . . . . 14 ⊢ (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31		elpreima 7049	. . . . . . . . . . . . . 14 ⊢ (𝑎 Fn ℕ0 → (𝑥 ∈ (◡𝑎 “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℕ0 ∧ (𝑎‘𝑥) ∈ (ℂ ∖ {0}))))
32	19, 30, 31	3syl 19	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 4753	. . . . . . . . . . . 12 ⊢ ((𝑎‘𝑥) ∈ (ℂ ∖ {0}) → (𝑎‘𝑥) ≠ 0)
35	33, 34	simpl2im 513	. . . . . . . . . . 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 500	. . . . . . . . . . . 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 26490	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛)))
38	18, 19, 37	syl2anc 596	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 3255	. . . . . . . . . . . 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 603	. . . . . . . . . . 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 16	. . . . . . . . . 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 3155	. . . . . . . . 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 5853	. . . . . . . . . 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 6053	. . . . . . . . 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 3322	. . . . . . . 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 418	. . . . . . 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 462	. . . . . 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 3162	. . . . 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 3176	. . . 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 16	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
52		ssrexv 4001	. . 3 ⊢ (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
53	29, 51, 52	mpsyl 69	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
54	28, 53	jca 521	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5650   “ cima 5654   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  ℂcc 11179  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186   ≤ cle 11325  ℕ₀cn0 12587  ℤcz 12674  ℤ_≥cuz 12946  ...cfz 13620  ↑cexp 14184  Σcsu 15833  Polycply 26482  coeffccoe 26484

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-rlim 15636  df-sum 15834  df-0p 25971  df-ply 26486  df-coe 26488

This theorem is used by:  coef  26529  dgrcl  26532  dgrub  26533  dgrlb  26535