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Theorem elply2 25557
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 25556 . . 3 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))))
2 simpr 485 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0))
3 simpll 765 . . . . . . . . . . . . . . . 16 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → 𝑆 ⊆ ℂ)
4 cnex 11132 . . . . . . . . . . . . . . . 16 ℂ ∈ V
5 ssexg 5280 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V)
63, 4, 5sylancl 586 . . . . . . . . . . . . . . 15 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → 𝑆 ∈ V)
7 snex 5388 . . . . . . . . . . . . . . 15 {0} ∈ V
8 unexg 7683 . . . . . . . . . . . . . . 15 ((𝑆 ∈ V ∧ {0} ∈ V) → (𝑆 ∪ {0}) ∈ V)
96, 7, 8sylancl 586 . . . . . . . . . . . . . 14 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → (𝑆 ∪ {0}) ∈ V)
10 nn0ex 12419 . . . . . . . . . . . . . 14 0 ∈ V
11 elmapg 8778 . . . . . . . . . . . . . 14 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑m0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
129, 10, 11sylancl 586 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → (𝑓 ∈ ((𝑆 ∪ {0}) ↑m0) ↔ 𝑓:ℕ0⟶(𝑆 ∪ {0})))
132, 12mpbid 231 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → 𝑓:ℕ0⟶(𝑆 ∪ {0}))
1413ffvelcdmda 7035 . . . . . . . . . . 11 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) ∧ 𝑥 ∈ ℕ0) → (𝑓𝑥) ∈ (𝑆 ∪ {0}))
15 ssun2 4133 . . . . . . . . . . . 12 {0} ⊆ (𝑆 ∪ {0})
16 c0ex 11149 . . . . . . . . . . . . 13 0 ∈ V
1716snss 4746 . . . . . . . . . . . 12 (0 ∈ (𝑆 ∪ {0}) ↔ {0} ⊆ (𝑆 ∪ {0}))
1815, 17mpbir 230 . . . . . . . . . . 11 0 ∈ (𝑆 ∪ {0})
19 ifcl 4531 . . . . . . . . . . 11 (((𝑓𝑥) ∈ (𝑆 ∪ {0}) ∧ 0 ∈ (𝑆 ∪ {0})) → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) ∈ (𝑆 ∪ {0}))
2014, 18, 19sylancl 586 . . . . . . . . . 10 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) ∧ 𝑥 ∈ ℕ0) → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) ∈ (𝑆 ∪ {0}))
2120fmpttd 7063 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶(𝑆 ∪ {0}))
22 elmapg 8778 . . . . . . . . . 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})))
239, 10, 22sylancl 586 . . . . . . . . 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})))
2421, 23mpbird 256 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∈ ((𝑆 ∪ {0}) ↑m0))
25 eleq1w 2820 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑘 → (𝑥 ∈ (0...𝑛) ↔ 𝑘 ∈ (0...𝑛)))
26 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑘 → (𝑓𝑥) = (𝑓𝑘))
2725, 26ifbieq1d 4510 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
28 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))
29 fvex 6855 . . . . . . . . . . . . . . . . 17 (𝑓𝑘) ∈ V
3029, 16ifex 4536 . . . . . . . . . . . . . . . 16 if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) ∈ V
3127, 28, 30fvmpt 6948 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0))
3231ad2antll 727 . . . . . . . . . . . . . 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 4495 . . . . . . . . . . . . . . 15 𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) = 0)
3433eqeq2d 2747 . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . 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 2960 . . . . . . . . . . . 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 13445 . . . . . . . . . . . 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 468 . . . . . . . . . 10 ((((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) ∧ 𝑘 ∈ ℕ0) → (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
4039ralrimiva 3143 . . . . . . . . 9 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → ∀𝑘 ∈ ℕ0 (((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) ≠ 0 → 𝑘𝑛))
41 simplr 767 . . . . . . . . . 10 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → 𝑛 ∈ ℕ0)
42 0cnd 11148 . . . . . . . . . . . . 13 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → 0 ∈ ℂ)
4342snssd 4769 . . . . . . . . . . . 12 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → {0} ⊆ ℂ)
443, 43unssd 4146 . . . . . . . . . . 11 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → (𝑆 ∪ {0}) ⊆ ℂ)
4521, 44fssd 6686 . . . . . . . . . 10 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)):ℕ0⟶ℂ)
46 plyco0 25553 . . . . . . . . . 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 584 . . . . . . . . 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 256 . . . . . . . 8 (((𝑆 ⊆ ℂ ∧ 𝑛 ∈ ℕ0) ∧ 𝑓 ∈ ((𝑆 ∪ {0}) ↑m0)) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) “ (ℤ‘(𝑛 + 1))) = {0})
49 eqidd 2737 . . . . . . . 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 6008 . . . . . . . . . . 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 2738 . . . . . . . . . 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 6841 . . . . . . . . . . . . . . 15 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → (𝑎𝑘) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘))
53 elfznn0 13534 . . . . . . . . . . . . . . . . 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 4492 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (0...𝑛) → if(𝑘 ∈ (0...𝑛), (𝑓𝑘), 0) = (𝑓𝑘))
5654, 55eqtrd 2776 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...𝑛) → ((𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0))‘𝑘) = (𝑓𝑘))
5752, 56sylan9eq 2796 . . . . . . . . . . . . . 14 ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎𝑘) = (𝑓𝑘))
5857oveq1d 7372 . . . . . . . . . . . . 13 ((𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑓𝑘) · (𝑧𝑘)))
5958sumeq2dv 15588 . . . . . . . . . . . 12 (𝑎 = (𝑥 ∈ ℕ0 ↦ if(𝑥 ∈ (0...𝑛), (𝑓𝑥), 0)) → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑓𝑘) · (𝑧𝑘)))
6059mpteq2dv 5207 . . . . . . . . . . 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 2747 . . . . . . . . . 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 631 . . . . . . . . 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 3581 . . . . . . . 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)))))
6424, 48, 49, 63syl12anc 835 . . . . . . 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 2740 . . . . . . . . 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 629 . . . . . . . 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 3175 . . . . . . 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 246 . . . . . 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 3152 . . . . 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 3165 . . . 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 569 . . 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 216 . 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 485 . . . . . 6 (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
7473reximi 3087 . . . . 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 3087 . . . 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 617 . . 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 25556 . . 3 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
7876, 77sylibr 233 . 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 208 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 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cun 3908  wss 3910  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188  cima 5636  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  cle 11190  0cn0 12413  cuz 12763  ...cfz 13424  cexp 13967  Σcsu 15570  Polycply 25545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-sum 15571  df-ply 25549
This theorem is referenced by:  plyadd  25578  plymul  25579  coeeu  25586  dgrlem  25590  coeid  25599
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