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Theorem dgrlem 26283
Description: Lemma for dgrcl 26287 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 26250 . . . 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 496 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
3 simplrr 778 . . . . . . 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 767 . . . . . . . . . . 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 26248 . . . . . . . . . . 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 11252 . . . . . . . . . . 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 4202 . . . . . . . . 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 11234 . . . . . . . . 9 ℂ ∈ V
11 ssexg 5329 . . . . . . . . 9 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
129, 10, 11sylancl 586 . . . . . . . 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 12530 . . . . . . . 8 0 ∈ V
14 elmapg 8878 . . . . . . . 8 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑m0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
1512, 13, 14sylancl 586 . . . . . . 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 232 . . . . . 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 777 . . . . . . . . 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 6754 . . . . . . . . 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 771 . . . . . . . . 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 773 . . . . . . . . 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 26281 . . . . . . . 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 2788 . . . . . . 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 6721 . . . . . 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 232 . . . . 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 412 . . . 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 3211 . . 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 12634 . . 3 0 ⊆ ℤ
30 ffn 6737 . . . . . . . . . . . . . 14 (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31 elpreima 7078 . . . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 503 . . . . . . . . . . 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 494 . . . . . . . . . . . 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 26246 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎𝑥) ≠ 0 → 𝑥𝑛)))
3818, 19, 37syl2anc 584 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 3249 . . . . . . . . . . . 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 591 . . . . . . . . . . 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 3144 . . . . . . . . 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 5889 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0))) ∧ ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))) → 𝑎 = 𝐴)
4544imaeq1d 6079 . . . . . . . . 9 (((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})))
4643, 45raleqtrdv 3326 . . . . . . . 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)
4746ex 412 . . . . . . 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))
4847expr 456 . . . . . 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)))
4948rexlimdv 3151 . . . . 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))
5049reximdva 3166 . . . 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))
512, 50mpd 15 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
52 ssrexv 4065 . . 3 (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
5329, 51, 52mpsyl 68 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
5428, 53jca 511 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 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  wss 3963  {csn 4631   class class class wbr 5148  cmpt 5231  ccnv 5688  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cc 11151  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  cle 11294  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  cexp 14099  Σcsu 15719  Polycply 26238  coeffccoe 26240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-0p 25719  df-ply 26242  df-coe 26244
This theorem is referenced by:  coef  26284  dgrcl  26287  dgrub  26288  dgrlb  26290
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