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Theorem elply2 23933
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 23932 . . 3 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))))
2 simpr 477 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0))
3 simpll 789 . . . . . . . . . . . . . . . 16 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑆 ⊆ ℂ)
4 cnex 10002 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5 ssexg 4795 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
63, 4, 5sylancl 693 . . . . . . . . . . . . . . 15 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑆 ∈ V)
7 snex 4899 . . . . . . . . . . . . . . 15 {0} ∈ V
8 unexg 6944 . . . . . . . . . . . . . . 15 ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
96, 7, 8sylancl 693 . . . . . . . . . . . . . 14 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑆 ∪ {0}) ∈ V)
10 nn0ex 11283 . . . . . . . . . . . . . 14 0 ∈ V
11 elmapg 7855 . . . . . . . . . . . . . 14 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
129, 10, 11sylancl 693 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
132, 12mpbid 222 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
1413ffvelrnda 6345 . . . . . . . . . . 11 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) ∧ 𝑥 ∈ ℕ0) → (𝑓𝑥) ∈ (𝑆 ∪ {0}))
15 ssun2 3769 . . . . . . . . . . . 12 {0} ⊆ (𝑆 ∪ {0})
16 c0ex 10019 . . . . . . . . . . . . 13 0 ∈ V
1716snss 4307 . . . . . . . . . . . 12 (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
1815, 17mpbir 221 . . . . . . . . . . 11 0 ∈ (𝑆 ∪ {0})
19 ifcl 4121 . . . . . . . . . . 11 (((𝑓𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) ∈ (𝑆 ∪ {0}))
2014, 18, 19sylancl 693 . . . . . . . . . 10 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) ∈ (𝑆 ∪ {0}))
21 eqid 2620 . . . . . . . . . 10 (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))
2220, 21fmptd 6371 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
23 elmapg 7855 . . . . . . . . . 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 693 . . . . . . . . 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 247 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑𝑚0))
26 eleq1 2687 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
27 fveq2 6178 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑘 → (𝑓𝑥) = (𝑓𝑘))
2826, 27ifbieq1d 4100 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
29 fvex 6188 . . . . . . . . . . . . . . . . 17 (𝑓𝑘) ∈ V
3029, 16ifex 4147 . . . . . . . . . . . . . . . 16 if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) ∈ V
3128, 21, 30fvmpt 6269 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
3231ad2antll 764 . . . . . . . . . . . . . 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 4086 . . . . . . . . . . . . . . 15 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) = 0)
3433eqeq2d 2630 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 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 2808 . . . . . . . . . . . 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 12330 . . . . . . . . . . . 12 (𝑘 ∈ (0...𝑛) → 𝑘𝑛)
3836, 37syl6 35 . . . . . . . . . . 11 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ (𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ∧ 𝑘 ∈ ℕ0)) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
3938anassrs 679 . . . . . . . . . 10 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) ∧ 𝑘 ∈ ℕ0) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
4039ralrimiva 2963 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
41 simplr 791 . . . . . . . . . 10 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 𝑛 ∈ ℕ0)
42 0cnd 10018 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → 0 ∈ ℂ)
4342snssd 4331 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → {0} ⊆ ℂ)
443, 43unssd 3781 . . . . . . . . . . 11 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑆 ∪ {0}) ⊆ ℂ)
4522, 44fssd 6044 . . . . . . . . . 10 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶ℂ)
46 plyco0 23929 . . . . . . . . . 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 692 . . . . . . . . 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 247 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑𝑚0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0})
49 eqidd 2621 . . . . . . . 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 5449 . . . . . . . . . . 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 2622 . . . . . . . . . 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 6177 . . . . . . . . . . . . . . 15 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → (𝑎𝑘) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘))
53 elfznn0 12417 . . . . . . . . . . . . . . . . 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 4083 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) = (𝑓𝑘))
5654, 55eqtrd 2654 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = (𝑓𝑘))
5752, 56sylan9eq 2674 . . . . . . . . . . . . . 14 ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) = (𝑓𝑘))
5857oveq1d 6650 . . . . . . . . . . . . 13 ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑓𝑘) · (𝑧𝑘)))
5958sumeq2dv 14414 . . . . . . . . . . . 12 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))
6059mpteq2dv 4736 . . . . . . . . . . 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 2630 . . . . . . . . . 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 746 . . . . . . . . 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 3304 . . . . . . . 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 1322 . . . . . . 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 2624 . . . . . . . . 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 739 . . . . . . . 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 3048 . . . . . . 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 237 . . . . . 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 3027 . . . . 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 3014 . . . 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 725 . . 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 207 . 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 477 . . . . . 6 (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
7473reximi 3008 . . . . 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 3008 . . . 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 592 . . 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 23932 . . 3 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
7876, 77sylibr 224 . 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 199 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 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  Vcvv 3195  cun 3565  wss 3567  ifcif 4077  {csn 4168   class class class wbr 4644  cmpt 4720  cima 5107  wf 5872  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  cc 9919  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  cle 10060  0cn0 11277  cuz 11672  ...cfz 12311  cexp 12843  Σcsu 14397  Polycply 23921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-seq 12785  df-sum 14398  df-ply 23925
This theorem is referenced by:  plyadd  23954  plymul  23955  coeeu  23962  dgrlem  23966  coeid  23975
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