Theorem elply2 26236

Description: The coefficient function can be assumed to have zeroes outside 0...𝑛. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
elply2	⊢ (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))))))

Distinct variable groups:   𝑘,𝑎,𝑛,𝑧,𝑆   𝐹,𝑎,𝑛

Allowed substitution hints:   𝐹(𝑧,𝑘)

Proof of Theorem elply2

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply 26235	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘)))))
2		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))
3		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → 𝑆 ⊆ ℂ)
4		cnex 11237	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
5		ssexg 5322	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
6	3, 4, 5	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → 𝑆 ∈ V)
7		snex 5435	. . . . . . . . . . . . . . 15 ⊢ {0} ∈ V
8		unexg 7764	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
9	6, 7, 8	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑆 ∪ {0}) ∈ V)
10		nn0ex 12534	. . . . . . . . . . . . . 14 ⊢ ℕ0 ∈ V
11		elmapg 8880	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
12	9, 10, 11	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
13	2, 12	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
14	13	ffvelcdmda 7103	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ 𝑥 ∈ ℕ0) → (𝑓‘𝑥) ∈ (𝑆 ∪ {0}))
15		ssun2 4178	. . . . . . . . . . . 12 ⊢ {0} ⊆ (𝑆 ∪ {0})
16		c0ex 11256	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
17	16	snss 4784	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
18	15, 17	mpbir 231	. . . . . . . . . . 11 ⊢ 0 ∈ (𝑆 ∪ {0})
19		ifcl 4570	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
20	14, 18, 19	sylancl 586	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
21	20	fmpttd 7134	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
22		elmapg 8880	. . . . . . . . . 10 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶(𝑆 ∪ {0})))
23	9, 10, 22	sylancl 586	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ↔ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶(𝑆 ∪ {0})))
24	21, 23	mpbird 257	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑m ℕ0))
25		eleq1w 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
26		fveq2 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
27	25, 26	ifbieq1d 4549	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
28		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))
29		fvex 6918	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑘) ∈ V
30	29, 16	ifex 4575	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) ∈ V
31	27, 28, 30	fvmpt 7015	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
32	31	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑘 ∈ ℕ0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
33		iffalse 4533	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = 0)
34	33	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 ∈ (0...𝑛) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) ↔ ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = 0))
35	32, 34	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑘 ∈ ℕ0)) → (¬ 𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = 0))
36	35	necon1ad 2956	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑘 ∈ ℕ0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑛)))
37		elfzle2 13569	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ≤ 𝑛)
38	36, 37	syl6 35	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑘 ∈ ℕ0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑛))
39	38	anassrs 467	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ 𝑘 ∈ ℕ0) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑛))
40	39	ralrimiva 3145	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑛))
41		simplr 768	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → 𝑛 ∈ ℕ0)
42		0cnd 11255	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → 0 ∈ ℂ)
43	42	snssd 4808	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → {0} ⊆ ℂ)
44	3, 43	unssd 4191	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑆 ∪ {0}) ⊆ ℂ)
45	21, 44	fssd 6752	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶ℂ)
46		plyco0 26232	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶ℂ) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑛)))
47	41, 45, 46	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ≤ 𝑛)))
48	40, 47	mpbird 257	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0})
49		eqidd 2737	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))))
50		imaeq1 6072	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))))
51	50	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0}))
52		fveq1 6904	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → (𝑎‘𝑘) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘))
53		elfznn0 13661	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
54	53, 31	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
55		iftrue 4530	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = (𝑓‘𝑘))
56	54, 55	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = (𝑓‘𝑘))
57	52, 56	sylan9eq 2796	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) = (𝑓‘𝑘))
58	57	oveq1d 7447	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑓‘𝑘) · (𝑧↑𝑘)))
59	58	sumeq2dv 15739	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘)))
60	59	mpteq2dv 5243	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))))
61	60	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘)))))
62	51, 61	anbi12d 632	. . . . . . . . 9 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))))))
63	62	rspcev 3621	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
64	24, 48, 49, 63	syl12anc 836	. . . . . . 7 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
65		eqeq1 2740	. . . . . . . . 9 ⊢ (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
66	65	anbi2d 630	. . . . . . . 8 ⊢ (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) → (((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
67	66	rexbidv 3178	. . . . . . 7 ⊢ (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) → (∃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 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
68	64, 67	syl5ibrcom 247	. . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
69	68	rexlimdva 3154	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) → (∃𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
70	69	reximdva 3167	. . . 4 ⊢ (𝑆 ⊆ ℂ → (∃𝑛 ∈ ℕ0 ∃𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘))) → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
71	70	imdistani 568	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑓 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓‘𝑘) · (𝑧↑𝑘)))) → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
72	1, 71	sylbi 217	. 2 ⊢ (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))))))
73		simpr 484	. . . . . 6 ⊢ (((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))))
74	73	reximi 3083	. . . . 5 ⊢ (∃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)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
75	74	reximi 3083	. . . 4 ⊢ (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) → ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
76	75	anim2i 617	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
77		elply 26235	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
78	76, 77	sylibr 234	. 2 ⊢ ((𝑆 ⊆ ℂ ∧ ∃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))
79	72, 78	impbii 209	1 ⊢ (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))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950  ifcif 4524  {csn 4625   class class class wbr 5142   ↦ cmpt 5224   “ cima 5687  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  ℂcc 11154  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161   ≤ cle 11297  ℕ₀cn0 12528  ℤ_≥cuz 12879  ...cfz 13548  ↑cexp 14103  Σcsu 15723  Polycply 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-seq 14044  df-sum 15724  df-ply 26228

This theorem is referenced by:  plyadd  26257  plymul  26258  coeeu  26265  dgrlem  26269  coeid  26278