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Theorem dgrlem 25967

Description: Lemma for dgrcl 25971 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 25934	. . . 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 776	. . . . . . 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 765	. . . . . . . . . . 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 25932	. . . . . . . . . . 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 11211	. . . . . . . . . . 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 4812	. . . . . . . . . 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 4186	. . . . . . . . 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 11193	. . . . . . . . 9 ⊢ ℂ ∈ V
11		ssexg 5323	. . . . . . . . 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 12482	. . . . . . . 8 ⊢ ℕ₀ ∈ V
14		elmapg 8835	. . . . . . . 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 775	. . . . . . . . 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 6735	. . . . . . . . 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 769	. . . . . . . . 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 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))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
22	4, 18, 19, 20, 21	coeeq 25965	. . . . . . . 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 6702	. . . . . 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 3211	. . 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 12585	. . 3 ⊢ ℕ₀ ⊆ ℤ
30		ffn 6717	. . . . . . . . . . . . . 14 ⊢ (𝑎:ℕ₀⟶ℂ → 𝑎 Fn ℕ₀)
31		elpreima 7059	. . . . . . . . . . . . . 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 4793	. . . . . . . . . . . 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 25930	. . . . . . . . . . . . . . 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 3248	. . . . . . . . . . . 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 3146	. . . . . . . . 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 5875	. . . . . . . . . . 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 6058	. . . . . . . . . 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 3325	. . . . . . . . 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 3153	. . . . 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 3168	. . . 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 4051	. . 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5675 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 ≤ cle 11253 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13488 ↑cexp 14031 Σcsu 15636 Polycply 25922 coeffccoe 25924

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-0p 25411 df-ply 25926 df-coe 25928

This theorem is referenced by: coef 25968 dgrcl 25971 dgrub 25972 dgrlb 25974

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