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Theorem elply2 23700
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 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
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.
StepHypRef Expression
1 elply 23699 . . 3 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))))
2 simpr 475 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0))
3 simpll 785 . . . . . . . . . . . . . . . 16 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑆 ⊆ ℂ)
4 cnex 9873 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5 ssexg 4726 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
63, 4, 5sylancl 692 . . . . . . . . . . . . . . 15 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑆 ∈ V)
7 snex 4829 . . . . . . . . . . . . . . 15 {0} ∈ V
8 unexg 6834 . . . . . . . . . . . . . . 15 ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
96, 7, 8sylancl 692 . . . . . . . . . . . . . 14 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑆 ∪ {0}) ∈ V)
10 nn0ex 11147 . . . . . . . . . . . . . 14 0 ∈ V
11 elmapg 7734 . . . . . . . . . . . . . 14 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
129, 10, 11sylancl 692 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
132, 12mpbid 220 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
1413ffvelrnda 6251 . . . . . . . . . . 11 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) ∧ 𝑥 ∈ ℕ0) → (𝑓𝑥) ∈ (𝑆 ∪ {0}))
15 ssun2 3738 . . . . . . . . . . . 12 {0} ⊆ (𝑆 ∪ {0})
16 c0ex 9890 . . . . . . . . . . . . 13 0 ∈ V
1716snss 4258 . . . . . . . . . . . 12 (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
1815, 17mpbir 219 . . . . . . . . . . 11 0 ∈ (𝑆 ∪ {0})
19 ifcl 4079 . . . . . . . . . . 11 (((𝑓𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) ∈ (𝑆 ∪ {0}))
2014, 18, 19sylancl 692 . . . . . . . . . 10 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) ∈ (𝑆 ∪ {0}))
21 eqid 2609 . . . . . . . . . 10 (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))
2220, 21fmptd 6276 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
23 elmapg 7734 . . . . . . . . . 10 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶(𝑆 ∪ {0})))
249, 10, 23sylancl 692 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶(𝑆 ∪ {0})))
2522, 24mpbird 245 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚0))
26 eleq1 2675 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
27 fveq2 6087 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑘 → (𝑓𝑥) = (𝑓𝑘))
2826, 27ifbieq1d 4058 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
29 fvex 6097 . . . . . . . . . . . . . . . . 17 (𝑓𝑘) ∈ V
3029, 16ifex 4105 . . . . . . . . . . . . . . . 16 if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) ∈ V
3128, 21, 30fvmpt 6175 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
3231ad2antll 760 . . . . . . . . . . . . . 14 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ∧ 𝑘 ∈ ℕ0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
33 iffalse 4044 . . . . . . . . . . . . . . 15 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) = 0)
3433eqeq2d 2619 . . . . . . . . . . . . . 14 𝑘 ∈ (0...𝑛) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) ↔ ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = 0))
3532, 34syl5ibcom 233 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ∧ 𝑘 ∈ ℕ0)) → (¬ 𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = 0))
3635necon1ad 2798 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ∧ 𝑘 ∈ ℕ0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘 ∈ (0...𝑛)))
37 elfzle2 12173 . . . . . . . . . . . 12 (𝑘 ∈ (0...𝑛) → 𝑘𝑛)
3836, 37syl6 34 . . . . . . . . . . 11 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ∧ 𝑘 ∈ ℕ0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
3938anassrs 677 . . . . . . . . . 10 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) ∧ 𝑘 ∈ ℕ0) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
4039ralrimiva 2948 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
41 simplr 787 . . . . . . . . . 10 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑛 ∈ ℕ0)
42 0cnd 9889 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 0 ∈ ℂ)
4342snssd 4280 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → {0} ⊆ ℂ)
443, 43unssd 3750 . . . . . . . . . . 11 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑆 ∪ {0}) ⊆ ℂ)
4522, 44fssd 5955 . . . . . . . . . 10 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶ℂ)
46 plyco0 23696 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶ℂ) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛)))
4741, 45, 46syl2anc 690 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛)))
4840, 47mpbird 245 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0})
49 eqidd 2610 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))))
50 imaeq1 5366 . . . . . . . . . . 11 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → (𝑎 “ (ℤ‘(𝑛 + 1))) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))))
5150eqeq1d 2611 . . . . . . . . . 10 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0}))
52 fveq1 6086 . . . . . . . . . . . . . . 15 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → (𝑎𝑘) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘))
53 elfznn0 12259 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
5453, 31syl 17 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
55 iftrue 4041 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) = (𝑓𝑘))
5654, 55eqtrd 2643 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = (𝑓𝑘))
5752, 56sylan9eq 2663 . . . . . . . . . . . . . 14 ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) = (𝑓𝑘))
5857oveq1d 6541 . . . . . . . . . . . . 13 ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑓𝑘) · (𝑧𝑘)))
5958sumeq2dv 14229 . . . . . . . . . . . 12 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))
6059mpteq2dv 4667 . . . . . . . . . . 11 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))))
6160eqeq2d 2619 . . . . . . . . . 10 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))))
6251, 61anbi12d 742 . . . . . . . . 9 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))))))
6362rspcev 3281 . . . . . . . 8 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚0) ∧ (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
6425, 48, 49, 63syl12anc 1315 . . . . . . 7 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
65 eqeq1 2613 . . . . . . . . 9 (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
6665anbi2d 735 . . . . . . . 8 (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
6766rexbidv 3033 . . . . . . 7 (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) → (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
6864, 67syl5ibrcom 235 . . . . . 6 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
6968rexlimdva 3012 . . . . 5 ((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) → (∃𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
7069reximdva 2999 . . . 4 (𝑆 ⊆ ℂ → (∃𝑛 ∈ ℕ0𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘))) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
7170imdistani 721 . . 3 ((𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))) → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
721, 71sylbi 205 . 2 (𝐹 ∈ (Poly‘𝑆) → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
73 simpr 475 . . . . . 6 (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
7473reximi 2993 . . . . 5 (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
7574reximi 2993 . . . 4 (∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
7675anim2i 590 . . 3 ((𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
77 elply 23699 . . 3 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
7876, 77sylibr 222 . 2 ((𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝐹 ∈ (Poly‘𝑆))
7972, 78impbii 197 1 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  cun 3537  wss 3539  ifcif 4035  {csn 4124   class class class wbr 4577  cmpt 4637  cima 5030  wf 5785  cfv 5789  (class class class)co 6526  𝑚 cmap 7721  cc 9790  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797  cle 9931  0cn0 11141  cuz 11521  ...cfz 12154  cexp 12679  Σcsu 14212  Polycply 23688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-n0 11142  df-z 11213  df-uz 11522  df-fz 12155  df-seq 12621  df-sum 14213  df-ply 23692
This theorem is referenced by:  plyadd  23721  plymul  23722  coeeu  23729  dgrlem  23733  coeid  23742
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