Theorem elply2 23756

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 23755	. . 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 476	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
3		simpll 786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑆 ⊆ ℂ)
4		cnex 9896	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
5		ssexg 4732	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
6	3, 4, 5	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑆 ∈ V)
7		snex 4835	. . . . . . . . . . . . . . 15 ⊢ {0} ∈ V
8		unexg 6857	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
9	6, 7, 8	sylancl 693	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑆 ∪ {0}) ∈ V)
10		nn0ex 11175	. . . . . . . . . . . . . 14 ⊢ ℕ0 ∈ V
11		elmapg 7757	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
12	9, 10, 11	sylancl 693	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
13	2, 12	mpbid 221	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
14	13	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → (𝑓‘𝑥) ∈ (𝑆 ∪ {0}))
15		ssun2 3739	. . . . . . . . . . . 12 ⊢ {0} ⊆ (𝑆 ∪ {0})
16		c0ex 9913	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
17	16	snss 4259	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
18	15, 17	mpbir 220	. . . . . . . . . . 11 ⊢ 0 ∈ (𝑆 ∪ {0})
19		ifcl 4080	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
20	14, 18, 19	sylancl 693	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
21		eqid 2610	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))
22	20, 21	fmptd 6292	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
23		elmapg 7757	. . . . . . . . . 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 693	. . . . . . . . 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 246	. . . . . . . 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 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
27		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
28	26, 27	ifbieq1d 4059	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
29		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑘) ∈ V
30	29, 16	ifex 4106	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) ∈ V
31	28, 21, 30	fvmpt 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
32	31	ad2antll 761	. . . . . . . . . . . . . 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 4045	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = 0)
34	33	eqeq2d 2620	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 2799	. . . . . . . . . . . 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 12216	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ≤ 𝑛)
38	36, 37	syl6 34	. . . . . . . . . . 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 678	. . . . . . . . . 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 2949	. . . . . . . . 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 788	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑛 ∈ ℕ0)
42		0cnd 9912	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 0 ∈ ℂ)
43	42	snssd 4281	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → {0} ⊆ ℂ)
44	3, 43	unssd 3751	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑆 ∪ {0}) ⊆ ℂ)
45	22, 44	fssd 5970	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶ℂ)
46		plyco0 23752	. . . . . . . . . 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 691	. . . . . . . . 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 246	. . . . . . . 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 2611	. . . . . . . 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 5380	. . . . . . . . . . 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 2612	. . . . . . . . . 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 6102	. . . . . . . . . . . . . . 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 12302	. . . . . . . . . . . . . . . . 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 4042	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = (𝑓‘𝑘))
56	54, 55	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = (𝑓‘𝑘))
57	52, 56	sylan9eq 2664	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) = (𝑓‘𝑘))
58	57	oveq1d 6564	. . . . . . . . . . . . 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 14281	. . . . . . . . . . . 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 4673	. . . . . . . . . . 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 2620	. . . . . . . . . 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 743	. . . . . . . . 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 3282	. . . . . . . 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 1316	. . . . . . 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 2614	. . . . . . . . 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 736	. . . . . . . 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 3034	. . . . . . 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 236	. . . . . 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 3013	. . . . 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 3000	. . . 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 722	. . 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 206	. 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 476	. . . . . 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 2994	. . . . 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 2994	. . . 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 591	. . 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 23755	. . 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 223	. 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 198	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-sum 14265  df-ply 23748

This theorem is referenced by:  plyadd  23777  plymul  23778  coeeu  23785  dgrlem  23789  coeid  23798