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Theorem dgrlem 23734
Description: Lemma for dgrcl 23738 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 23701 . . . 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 478 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
3 simplrr 796 . . . . . . 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 785 . . . . . . . . . . 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 23699 . . . . . . . . . . 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 9890 . . . . . . . . . . 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 4280 . . . . . . . . . 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 3750 . . . . . . . . 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 9874 . . . . . . . . 9 ℂ ∈ V
11 ssexg 4727 . . . . . . . . 9 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
129, 10, 11sylancl 692 . . . . . . . 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 11148 . . . . . . . 8 0 ∈ V
14 elmapg 7735 . . . . . . . 8 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
1512, 13, 14sylancl 692 . . . . . . 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 220 . . . . . 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 795 . . . . . . . . 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 5956 . . . . . . . . 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 789 . . . . . . . . 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 791 . . . . . . . . 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 23732 . . . . . . . 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, 22syl5req 2656 . . . . . . 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 5929 . . . . . 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 220 . . . . 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 448 . . . 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 3019 . . 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 11234 . . 3 0 ⊆ ℤ
30 ffn 5944 . . . . . . . . . . . . . . 15 (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31 elpreima 6230 . . . . . . . . . . . . . . 15 (𝑎 Fn ℕ0 → (𝑥 ∈ (𝑎 “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℕ0 ∧ (𝑎𝑥) ∈ (ℂ ∖ {0}))))
3219, 30, 313syl 18 . . . . . . . . . . . . . 14 (((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 499 . . . . . . . . . . . . 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})))
3433simprd 477 . . . . . . . . . . . 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}))) → (𝑎𝑥) ∈ (ℂ ∖ {0}))
35 eldifsni 4260 . . . . . . . . . . . 12 ((𝑎𝑥) ∈ (ℂ ∖ {0}) → (𝑎𝑥) ≠ 0)
3634, 35syl 17 . . . . . . . . . . 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)
3733simpld 473 . . . . . . . . . . . 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)
38 plyco0 23697 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎𝑥) ≠ 0 → 𝑥𝑛)))
3918, 19, 38syl2anc 690 . . . . . . . . . . . . . 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)))
4020, 39mpbid 220 . . . . . . . . . . . . 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))
4140r19.21bi 2915 . . . . . . . . . . . 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))
4237, 41syldan 485 . . . . . . . . . . 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))
4336, 42mpd 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)
4443ralrimiva 2948 . . . . . . . . 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)
4523cnveqd 5208 . . . . . . . . . . 11 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎 = 𝐴)
4645imaeq1d 5371 . . . . . . . . . 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})))
4746raleqdv 3120 . . . . . . . . 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))
4844, 47mpbid 220 . . . . . . . 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)
4948ex 448 . . . . . . 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))
5049expr 640 . . . . . 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)))
5150rexlimdv 3011 . . . . 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))
5251reximdva 2999 . . . 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))
532, 52mpd 15 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
54 ssrexv 3629 . . 3 (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
5529, 53, 54mpsyl 65 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
5628, 55jca 552 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 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  cun 3537  wss 3539  {csn 4124   class class class wbr 4577  cmpt 4637  ccnv 5027  cima 5031   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7722  cc 9791  0cc0 9793  1c1 9794   + caddc 9796   · cmul 9798  cle 9932  0cn0 11142  cz 11213  cuz 11522  ...cfz 12155  cexp 12680  Σcsu 14213  Polycply 23689  coeffccoe 23691
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 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871  ax-addf 9872
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-rmo 2903  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-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-fz 12156  df-fzo 12293  df-fl 12413  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-rlim 14017  df-sum 14214  df-0p 23188  df-ply 23693  df-coe 23695
This theorem is referenced by:  coef  23735  dgrcl  23738  dgrub  23739  dgrlb  23741
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