Theorem elply2 14971

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:   𝑆,𝑎,𝑛   𝑘,𝑎,𝑧,𝑛   𝐹,𝑎,𝑛

Allowed substitution hints:   𝑆(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem elply2

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

Step	Hyp	Ref	Expression
1		elply 14970	. . 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 110	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
3		simpll 527	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑆 ⊆ ℂ)
4		cnex 8003	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
5		ssexg 4172	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
6	3, 4, 5	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑆 ∈ V)
7		c0ex 8020	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
8	7	snex 4218	. . . . . . . . . . . . . . 15 ⊢ {0} ∈ V
9		unexg 4478	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
10	6, 8, 9	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑆 ∪ {0}) ∈ V)
11		nn0ex 9255	. . . . . . . . . . . . . 14 ⊢ ℕ0 ∈ V
12		elmapg 6720	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
13	10, 11, 12	sylancl 413	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
14	2, 13	mpbid 147	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
15	14	ffvelcdmda 5697	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → (𝑓‘𝑥) ∈ (𝑆 ∪ {0}))
16		ssun2 3327	. . . . . . . . . . . . 13 ⊢ {0} ⊆ (𝑆 ∪ {0})
17	7	snss 3757	. . . . . . . . . . . . 13 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
18	16, 17	mpbir 146	. . . . . . . . . . . 12 ⊢ 0 ∈ (𝑆 ∪ {0})
19	18	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → 0 ∈ (𝑆 ∪ {0}))
20		nn0z 9346	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ)
21	20	adantl 277	. . . . . . . . . . . 12 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℤ)
22		0zd 9338	. . . . . . . . . . . 12 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → 0 ∈ ℤ)
23		simpllr 534	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → 𝑛 ∈ ℕ0)
24	23	nn0zd 9446	. . . . . . . . . . . 12 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → 𝑛 ∈ ℤ)
25		fzdcel 10115	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑛 ∈ ℤ) → DECID 𝑥 ∈ (0...𝑛))
26	21, 22, 24, 25	syl3anc 1249	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → DECID 𝑥 ∈ (0...𝑛))
27	15, 19, 26	ifcldcd 3597	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
28	27	fmpttd 5717	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
29		elmapg 6720	. . . . . . . . . 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})))
30	10, 11, 29	sylancl 413	. . . . . . . . 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})))
31	28, 30	mpbird 167	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0))
32		mptima 5021	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = ran (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))
33		fznuz 10177	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (0...𝑛) → ¬ 𝑥 ∈ (ℤ≥‘(𝑛 + 1)))
34		elinel2 3350	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) → 𝑥 ∈ (ℤ≥‘(𝑛 + 1)))
35	33, 34	nsyl3 627	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑥 ∈ (0...𝑛))
36	35	iffalsed 3571	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) = 0)
37	36	mpteq2ia 4119	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ 0)
38		fconstmpt 4710	. . . . . . . . . . . . 13 ⊢ ((ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) × {0}) = (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ 0)
39	37, 38	eqtr4i 2220	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = ((ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) × {0})
40	39	rneqi 4894	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = ran ((ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) × {0})
41		peano2nn0 9289	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
42		nn0z 9346	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
43	42	peano2zd 9451	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℤ)
44	43	uzidd 9616	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)))
45	41, 44	elind 3348	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))))
46		elex2 2779	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) → ∃𝑤 𝑤 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))))
47		rnxpm 5099	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) → ran ((ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) × {0}) = {0})
48	45, 46, 47	3syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 → ran ((ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) × {0}) = {0})
49	40, 48	eqtrid 2241	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ran (𝑥 ∈ (ℕ0 ∩ (ℤ≥‘(𝑛 + 1))) ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = {0})
50	32, 49	eqtrid 2241	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0})
51	50	ad2antlr 489	. . . . . . . 8 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) “ (ℤ≥‘(𝑛 + 1))) = {0})
52		eqidd 2197	. . . . . . . 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))))
53		imaeq1 5004	. . . . . . . . . . 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))))
54	53	eqeq1d 2205	. . . . . . . . . 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}))
55		fveq1 5557	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) → (𝑎‘𝑘) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘))
56		elfznn0 10189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
57		eleq1w 2257	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
58		fveq2 5558	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
59	57, 58	ifbieq1d 3583	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
60		eqid 2196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))
61		vex 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑓 ∈ V
62		vex 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑘 ∈ V
63	61, 62	fvex 5578	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑘) ∈ V
64	63, 7	ifex 4521	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) ∈ V
65	59, 60, 64	fvmpt 5638	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
66	56, 65	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
67		iftrue 3566	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = (𝑓‘𝑘))
68	66, 67	eqtrd 2229	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = (𝑓‘𝑘))
69	55, 68	sylan9eq 2249	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) = (𝑓‘𝑘))
70	69	oveq1d 5937	. . . . . . . . . . . . 13 ⊢ ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑓‘𝑘) · (𝑧↑𝑘)))
71	70	sumeq2dv 11533	. . . . . . . . . . . 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)))
72	71	mpteq2dv 4124	. . . . . . . . . . 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))))
73	72	eqeq2d 2208	. . . . . . . . . 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)))))
74	54, 73	anbi12d 473	. . . . . . . . 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))))))
75	74	rspcev 2868	. . . . . . . 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)))))
76	31, 51, 52, 75	syl12anc 1247	. . . . . . 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)))))
77		eqeq1 2203	. . . . . . . . 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)))))
78	77	anbi2d 464	. . . . . . . 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))))))
79	78	rexbidv 2498	. . . . . . 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))))))
80	76, 79	syl5ibrcom 157	. . . . . 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))))))
81	80	rexlimdva 2614	. . . . 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))))))
82	81	reximdva 2599	. . . 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))))))
83	82	imdistani 445	. . 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))))))
84	1, 83	sylbi 121	. 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))))))
85		simpr 110	. . . . . 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))))
86	85	reximi 2594	. . . . 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))))
87	86	reximi 2594	. . . 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))))
88	87	anim2i 342	. . 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)))))
89		elply 14970	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
90	88, 89	sylibr 134	. 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))
91	84, 90	impbii 126	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:   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157  ifcif 3561  {csn 3622   ↦ cmpt 4094   × cxp 4661  ran crn 4664   “ cima 4666  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ↑_𝑚 cmap 6707  ℂcc 7877  0cc0 7879  1c1 7880   + caddc 7882   · cmul 7884  ℕ₀cn0 9249  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083  ↑cexp 10630  Σcsu 11518  Polycply 14964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540  df-sumdc 11519  df-ply 14966

This theorem is referenced by:  plyadd  14987  plymul  14988  plyco  14995  dvply2g  15002