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Theorem dgrlem 26208
Description: Lemma for dgrcl 26212 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypothesis
Ref Expression
dgrval.1 𝐴 = (coeff‘𝐹)
Assertion
Ref Expression
dgrlem (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
Distinct variable groups:   𝑥,𝑛,𝐴   𝑛,𝐹,𝑥   𝑆,𝑛,𝑥

Proof of Theorem dgrlem
Dummy variables 𝑎 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elply2 26175 . . . 4 (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
21simprbi 495 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
3 simplrr 776 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))
4 simpll 765 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝐹 ∈ (Poly‘𝑆))
5 plybss 26173 . . . . . . . . . . 11 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
64, 5syl 17 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑆 ⊆ ℂ)
7 0cnd 11239 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 0 ∈ ℂ)
87snssd 4814 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → {0} ⊆ ℂ)
96, 8unssd 4184 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑆 ∪ {0}) ⊆ ℂ)
10 cnex 11221 . . . . . . . . 9 ℂ ∈ V
11 ssexg 5324 . . . . . . . . 9 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
129, 10, 11sylancl 584 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑆 ∪ {0}) ∈ V)
13 nn0ex 12511 . . . . . . . 8 0 ∈ V
14 elmapg 8858 . . . . . . . 8 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
1512, 13, 14sylancl 584 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
163, 15mpbid 231 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎:ℕ0⟶(𝑆 ∪ {0}))
17 dgrval.1 . . . . . . . 8 𝐴 = (coeff‘𝐹)
18 simplrl 775 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑛 ∈ ℕ0)
1916, 9fssd 6740 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎:ℕ0⟶ℂ)
20 simprl 769 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑎 “ (ℤ‘(𝑛 + 1))) = {0})
21 simprr 771 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
224, 18, 19, 20, 21coeeq 26206 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (coeff‘𝐹) = 𝑎)
2317, 22eqtr2id 2778 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎 = 𝐴)
2423feq1d 6708 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑎:ℕ0⟶(𝑆 ∪ {0}) ↔ 𝐴:ℕ0⟶(𝑆 ∪ {0})))
2516, 24mpbid 231 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
2625ex 411 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → 𝐴:ℕ0⟶(𝑆 ∪ {0})))
2726rexlimdvva 3201 . . 3 (𝐹 ∈ (Poly‘𝑆) → (∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → 𝐴:ℕ0⟶(𝑆 ∪ {0})))
282, 27mpd 15 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
29 nn0ssz 12614 . . 3 0 ⊆ ℤ
30 ffn 6723 . . . . . . . . . . . . . 14 (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31 elpreima 7066 . . . . . . . . . . . . . 14 (𝑎 Fn ℕ0 → (𝑥 ∈ (𝑎 “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℕ0 ∧ (𝑎𝑥) ∈ (ℂ ∖ {0}))))
3219, 30, 313syl 18 . . . . . . . . . . . . 13 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑥 ∈ (𝑎 “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℕ0 ∧ (𝑎𝑥) ∈ (ℂ ∖ {0}))))
3332biimpa 475 . . . . . . . . . . . 12 ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) ∧ 𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))) → (𝑥 ∈ ℕ0 ∧ (𝑎𝑥) ∈ (ℂ ∖ {0})))
34 eldifsni 4795 . . . . . . . . . . . 12 ((𝑎𝑥) ∈ (ℂ ∖ {0}) → (𝑎𝑥) ≠ 0)
3533, 34simpl2im 502 . . . . . . . . . . 11 ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) ∧ 𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))) → (𝑎𝑥) ≠ 0)
3633simpld 493 . . . . . . . . . . . 12 ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) ∧ 𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))) → 𝑥 ∈ ℕ0)
37 plyco0 26171 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎𝑥) ≠ 0 → 𝑥𝑛)))
3818, 19, 37syl2anc 582 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎𝑥) ≠ 0 → 𝑥𝑛)))
3920, 38mpbid 231 . . . . . . . . . . . . 13 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → ∀𝑥 ∈ ℕ0 ((𝑎𝑥) ≠ 0 → 𝑥𝑛))
4039r19.21bi 3238 . . . . . . . . . . . 12 ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) ∧ 𝑥 ∈ ℕ0) → ((𝑎𝑥) ≠ 0 → 𝑥𝑛))
4136, 40syldan 589 . . . . . . . . . . 11 ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) ∧ 𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))) → ((𝑎𝑥) ≠ 0 → 𝑥𝑛))
4235, 41mpd 15 . . . . . . . . . 10 ((((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) ∧ 𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))) → 𝑥𝑛)
4342ralrimiva 3135 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → ∀𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))𝑥𝑛)
4423cnveqd 5878 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎 = 𝐴)
4544imaeq1d 6063 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (𝑎 “ (ℂ ∖ {0})) = (𝐴 “ (ℂ ∖ {0})))
4645raleqdv 3314 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → (∀𝑥 ∈ (𝑎 “ (ℂ ∖ {0}))𝑥𝑛 ↔ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
4743, 46mpbid 231 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
4847ex 411 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
4948expr 455 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)))
5049rexlimdv 3142 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
5150reximdva 3157 . . . 4 (𝐹 ∈ (Poly‘𝑆) → (∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) → ∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
522, 51mpd 15 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
53 ssrexv 4046 . . 3 (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
5429, 52, 53mpsyl 68 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
5528, 54jca 510 1 (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  Vcvv 3461  cdif 3941  cun 3942  wss 3944  {csn 4630   class class class wbr 5149  cmpt 5232  ccnv 5677  cima 5681   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  m cmap 8845  cc 11138  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145  cle 11281  0cn0 12505  cz 12591  cuz 12855  ...cfz 13519  cexp 14062  Σcsu 15668  Polycply 26163  coeffccoe 26165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-rlim 15469  df-sum 15669  df-0p 25643  df-ply 26167  df-coe 26169
This theorem is referenced by:  coef  26209  dgrcl  26212  dgrub  26213  dgrlb  26215
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