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Theorem dgrlem 25743

Description: Lemma for dgrcl 25747 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 25710	. . . 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 498	. . 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 777	. . . . . . 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 766	. . . . . . . . . . 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 25708	. . . . . . . . . . 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 11207	. . . . . . . . . . 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 4813	. . . . . . . . . 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 4187	. . . . . . . . 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 11191	. . . . . . . . 9 ⊢ ℂ ∈ V
11		ssexg 5324	. . . . . . . . 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 12478	. . . . . . . 8 ⊢ ℕ₀ ∈ V
14		elmapg 8833	. . . . . . . 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 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 776	. . . . . . . . 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 6736	. . . . . . . . 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 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))))) → (𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0})
21		simprr 772	. . . . . . . . 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 25741	. . . . . . . 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 2786	. . . . . . 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 6703	. . . . . 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 414	. . . 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 3212	. . 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 12581	. . 3 ⊢ ℕ₀ ⊆ ℤ
30		ffn 6718	. . . . . . . . . . . . . 14 ⊢ (𝑎:ℕ₀⟶ℂ → 𝑎 Fn ℕ₀)
31		elpreima 7060	. . . . . . . . . . . . . 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 478	. . . . . . . . . . . 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 4794	. . . . . . . . . . . 12 ⊢ ((𝑎‘𝑥) ∈ (ℂ ∖ {0}) → (𝑎‘𝑥) ≠ 0)
35	33, 34	simpl2im 505	. . . . . . . . . . 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 496	. . . . . . . . . . . 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 25706	. . . . . . . . . . . . . . 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 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 3249	. . . . . . . . . . . 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 3147	. . . . . . . . 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 5876	. . . . . . . . . . 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 6059	. . . . . . . . . 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 3326	. . . . . . . . 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 414	. . . . . . 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 458	. . . . . 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 3154	. . . . 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 3169	. . . 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 4052	. . 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 513	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ₀⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (^◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 {csn 4629 class class class wbr 5149 ↦ cmpt 5232 ^◡ccnv 5676 “ cima 5680 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ≤ cle 11249 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 ↑cexp 14027 Σcsu 15632 Polycply 25698 coeffccoe 25700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-0p 25187 df-ply 25702 df-coe 25704

This theorem is referenced by: coef 25744 dgrcl 25747 dgrub 25748 dgrlb 25750

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