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Theorem elply2 25944

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 25943	. . 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 483	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀))
3		simpll 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → 𝑆 ⊆ ℂ)
4		cnex 11195	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
5		ssexg 5324	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
6	3, 4, 5	sylancl 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → 𝑆 ∈ V)
7		snex 5432	. . . . . . . . . . . . . . 15 ⊢ {0} ∈ V
8		unexg 7740	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
9	6, 7, 8	sylancl 584	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → (𝑆 ∪ {0}) ∈ V)
10		nn0ex 12484	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
11		elmapg 8837	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∪ {0}) ∈ V ∧ ℕ₀ ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀) ↔ 𝑓:ℕ₀⟶(𝑆 ∪ {0})))
12	9, 10, 11	sylancl 584	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀) ↔ 𝑓:ℕ₀⟶(𝑆 ∪ {0})))
13	2, 12	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → 𝑓:ℕ₀⟶(𝑆 ∪ {0}))
14	13	ffvelcdmda 7087	. . . . . . . . . . 11 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) ∧ 𝑥 ∈ ℕ₀) → (𝑓‘𝑥) ∈ (𝑆 ∪ {0}))
15		ssun2 4174	. . . . . . . . . . . 12 ⊢ {0} ⊆ (𝑆 ∪ {0})
16		c0ex 11214	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
17	16	snss 4790	. . . . . . . . . . . 12 ⊢ (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
18	15, 17	mpbir 230	. . . . . . . . . . 11 ⊢ 0 ∈ (𝑆 ∪ {0})
19		ifcl 4574	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
20	14, 18, 19	sylancl 584	. . . . . . . . . 10 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) ∧ 𝑥 ∈ ℕ₀) → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) ∈ (𝑆 ∪ {0}))
21	20	fmpttd 7117	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → (𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ₀⟶(𝑆 ∪ {0}))
22		elmapg 8837	. . . . . . . . . 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 584	. . . . . . . . 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 256	. . . . . . . 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 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
26		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑓‘𝑥) = (𝑓‘𝑘))
27	25, 26	ifbieq1d 4553	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
28		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) = (𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))
29		fvex 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓‘𝑘) ∈ V
30	29, 16	ifex 4579	. . . . . . . . . . . . . . . 16 ⊢ if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) ∈ V
31	27, 28, 30	fvmpt 6999	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → ((𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0))
32	31	ad2antll 725	. . . . . . . . . . . . . 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 4538	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = 0)
34	33	eqeq2d 2741	. . . . . . . . . . . . . 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 244	. . . . . . . . . . . . 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 2955	. . . . . . . . . . . 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 13511	. . . . . . . . . . . 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 466	. . . . . . . . . 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 3144	. . . . . . . . 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 765	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → 𝑛 ∈ ℕ₀)
42		0cnd 11213	. . . . . . . . . . . . 13 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → 0 ∈ ℂ)
43	42	snssd 4813	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → {0} ⊆ ℂ)
44	3, 43	unssd 4187	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → (𝑆 ∪ {0}) ⊆ ℂ)
45	21, 44	fssd 6736	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑_m ℕ₀)) → (𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)):ℕ₀⟶ℂ)
46		plyco0 25940	. . . . . . . . . 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 582	. . . . . . . . 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 256	. . . . . . . 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 2731	. . . . . . . 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 6055	. . . . . . . . . . 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 2732	. . . . . . . . . 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 6891	. . . . . . . . . . . . . . 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 13600	. . . . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓‘𝑘), 0) = (𝑓‘𝑘))
56	54, 55	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0))‘𝑘) = (𝑓‘𝑘))
57	52, 56	sylan9eq 2790	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 ∈ (0...𝑛), (𝑓‘𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) = (𝑓‘𝑘))
58	57	oveq1d 7428	. . . . . . . . . . . . 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 15655	. . . . . . . . . . . 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 5251	. . . . . . . . . . 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 2741	. . . . . . . . . 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 629	. . . . . . . . 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 3613	. . . . . . . 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 833	. . . . . . 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 2734	. . . . . . . . 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 627	. . . . . . . 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 3176	. . . . . . 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 246	. . . . . 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 3153	. . . . 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 3166	. . . 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 567	. . 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 216	. 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 483	. . . . . 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 3082	. . . . 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 3082	. . . 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 615	. . 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 25943	. . 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 233	. 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 208	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))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ∪ cun 3947 ⊆ wss 3949 ifcif 4529 {csn 4629 class class class wbr 5149 ↦ cmpt 5232 “ cima 5680 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 ↑_m cmap 8824 ℂcc 11112 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 ≤ cle 11255 ℕ₀cn0 12478 ℤ_≥cuz 12828 ...cfz 13490 ↑cexp 14033 Σcsu 15638 Polycply 25932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-seq 13973 df-sum 15639 df-ply 25936

This theorem is referenced by: plyadd 25965 plymul 25966 coeeu 25973 dgrlem 25977 coeid 25986

W3C validator