Theorem dgrlem 25386

Description: Lemma for dgrcl 25390 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 25353	. . . 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 775	. . . . . . 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 764	. . . . . . . . . . 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 25351	. . . . . . . . . . 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 10967	. . . . . . . . . . 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 4748	. . . . . . . . . 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 4125	. . . . . . . . 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 10951	. . . . . . . . 9 ⊢ ℂ ∈ V
11		ssexg 5251	. . . . . . . . 9 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
12	9, 10, 11	sylancl 586	. . . . . . . 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 12237	. . . . . . . 8 ⊢ ℕ0 ∈ V
14		elmapg 8609	. . . . . . . 8 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
15	12, 13, 14	sylancl 586	. . . . . . 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 231	. . . . . 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 774	. . . . . . . . 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 6615	. . . . . . . . 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 768	. . . . . . . . 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 770	. . . . . . . . 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 25384	. . . . . . . 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 2793	. . . . . . 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 6582	. . . . . 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 231	. . . . 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 413	. . . 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 3225	. . 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 12339	. . 3 ⊢ ℕ0 ⊆ ℤ
30		ffn 6597	. . . . . . . . . . . . . 14 ⊢ (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31		elpreima 6930	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 4729	. . . . . . . . . . . 12 ⊢ ((𝑎‘𝑥) ∈ (ℂ ∖ {0}) → (𝑎‘𝑥) ≠ 0)
35	33, 34	simpl2im 504	. . . . . . . . . . 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 495	. . . . . . . . . . . 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 25349	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎‘𝑥) ≠ 0 → 𝑥 ≤ 𝑛)))
38	18, 19, 37	syl2anc 584	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 3135	. . . . . . . . . . . 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 591	. . . . . . . . . . 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 3110	. . . . . . . . 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 5782	. . . . . . . . . . 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))))) → ◡𝑎 = ◡𝐴)
45	44	imaeq1d 5966	. . . . . . . . . 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 “ (ℂ ∖ {0})) = (◡𝐴 “ (ℂ ∖ {0})))
46	45	raleqdv 3347	. . . . . . . . 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 ↔ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
47	43, 46	mpbid 231	. . . . . . . 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)
48	47	ex 413	. . . . . . 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))
49	48	expr 457	. . . . . 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)))
50	49	rexlimdv 3214	. . . . 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))
51	50	reximdva 3205	. . . 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))
52	2, 51	mpd 15	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
53		ssrexv 3993	. . 3 ⊢ (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0 ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))
54	29, 52, 53	mpsyl 68	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)
55	28, 54	jca 512	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ≠ wne 2945  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ∖ cdif 3889   ∪ cun 3890   ⊆ wss 3892  {csn 4567   class class class wbr 5079   ↦ cmpt 5162  ^◡ccnv 5588   “ cima 5592   Fn wfn 6426  ⟶wf 6427  ‘cfv 6431  (class class class)co 7269   ↑_m cmap 8596  ℂcc 10868  0cc0 10870  1c1 10871   + caddc 10873   · cmul 10875   ≤ cle 11009  ℕ₀cn0 12231  ℤcz 12317  ℤ_≥cuz 12579  ...cfz 13236  ↑cexp 13778  Σcsu 15393  Polycply 25341  coeffccoe 25343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-inf2 9375  ax-cnex 10926  ax-resscn 10927  ax-1cn 10928  ax-icn 10929  ax-addcl 10930  ax-addrcl 10931  ax-mulcl 10932  ax-mulrcl 10933  ax-mulcom 10934  ax-addass 10935  ax-mulass 10936  ax-distr 10937  ax-i2m1 10938  ax-1ne0 10939  ax-1rid 10940  ax-rnegex 10941  ax-rrecex 10942  ax-cnre 10943  ax-pre-lttri 10944  ax-pre-lttrn 10945  ax-pre-ltadd 10946  ax-pre-mulgt0 10947  ax-pre-sup 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-isom 6440  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-of 7525  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-er 8479  df-map 8598  df-pm 8599  df-en 8715  df-dom 8716  df-sdom 8717  df-fin 8718  df-sup 9177  df-inf 9178  df-oi 9245  df-card 9696  df-pnf 11010  df-mnf 11011  df-xr 11012  df-ltxr 11013  df-le 11014  df-sub 11205  df-neg 11206  df-div 11631  df-nn 11972  df-2 12034  df-3 12035  df-n0 12232  df-z 12318  df-uz 12580  df-rp 12728  df-fz 13237  df-fzo 13380  df-fl 13508  df-seq 13718  df-exp 13779  df-hash 14041  df-cj 14806  df-re 14807  df-im 14808  df-sqrt 14942  df-abs 14943  df-clim 15193  df-rlim 15194  df-sum 15394  df-0p 24830  df-ply 25345  df-coe 25347

This theorem is referenced by:  coef  25387  dgrcl  25390  dgrub  25391  dgrlb  25393