Theorem elply2 23952

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 23951	. . 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 477	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
3		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑆 ⊆ ℂ)
4		cnex 10017	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
5		ssexg 4804	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
6	3, 4, 5	sylancl 694	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑆 ∈ V)
7		snex 4908	. . . . . . . . . . . . . . 15 ⊢ {0} ∈ V
8		unexg 6959	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
9	6, 7, 8	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑆 ∪ {0}) ∈ V)
10		nn0ex 11298	. . . . . . . . . . . . . 14 ⊢ ℕ0 ∈ V
11		elmapg 7870	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
12	9, 10, 11	sylancl 694	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
13	2, 12	mpbid 222	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
14	13	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → (𝑓‘𝑥) ∈ (𝑆 ∪ {0}))
15		ssun2 3777	. . . . . . . . . . . 12 ⊢ {0} ⊆ (𝑆 ∪ {0})
16		c0ex 10034	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
17	16	snss 4316	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
18	15, 17	mpbir 221	. . . . . . . . . . 11 ⊢ 0 ∈ (𝑆 ∪ {0})
19		ifcl 4130	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
20	14, 18, 19	sylancl 694	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
21		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))
22	20, 21	fmptd 6385	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
23		elmapg 7870	. . . . . . . . . 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})))
24	9, 10, 23	sylancl 694	. . . . . . . . 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})))
25	22, 24	mpbird 247	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
26		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
27		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
28	26, 27	ifbieq1d 4109	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
29		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑘) ∈ V
30	29, 16	ifex 4156	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) ∈ V
31	28, 21, 30	fvmpt 6282	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
32	31	ad2antll 765	. . . . . . . . . . . . . 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 4095	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = 0)
34	33	eqeq2d 2632	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 2811	. . . . . . . . . . . 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 12345	. . . . . . . . . . . 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 680	. . . . . . . . . 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 2966	. . . . . . . . 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 792	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑛 ∈ ℕ0)
42		0cnd 10033	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 0 ∈ ℂ)
43	42	snssd 4340	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → {0} ⊆ ℂ)
44	3, 43	unssd 3789	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑆 ∪ {0}) ⊆ ℂ)
45	22, 44	fssd 6057	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶ℂ)
46		plyco0 23948	. . . . . . . . . 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 693	. . . . . . . . 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 247	. . . . . . . 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 2623	. . . . . . . 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 5461	. . . . . . . . . . 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 2624	. . . . . . . . . 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 6190	. . . . . . . . . . . . . . 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 12433	. . . . . . . . . . . . . . . . 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 4092	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = (𝑓‘𝑘))
56	54, 55	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = (𝑓‘𝑘))
57	52, 56	sylan9eq 2676	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) = (𝑓‘𝑘))
58	57	oveq1d 6665	. . . . . . . . . . . . 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 14433	. . . . . . . . . . . 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 4745	. . . . . . . . . . 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 2632	. . . . . . . . . 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 747	. . . . . . . . 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 3309	. . . . . . . 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	25, 48, 49, 63	syl12anc 1324	. . . . . . 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 2626	. . . . . . . . 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 740	. . . . . . . 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 3052	. . . . . . 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 237	. . . . . 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 3031	. . . . 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 3017	. . . 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 726	. . 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 207	. 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 477	. . . . . 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 3011	. . . . 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 3011	. . . 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 593	. . 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 23951	. . 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 224	. 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 199	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ifcif 4086  {csn 4177   class class class wbr 4653   ↦ cmpt 4729   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   ≤ cle 10075  ℕ₀cn0 11292  ℤ_≥cuz 11687  ...cfz 12326  ↑cexp 12860  Σcsu 14416  Polycply 23940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-sum 14417  df-ply 23944

This theorem is referenced by:  plyadd  23973  plymul  23974  coeeu  23981  dgrlem  23985  coeid  23994