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Theorem dgrlem 24030
Description: Lemma for dgrcl 24034 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 23997 . . . 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 479 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
3 simplrr 818 . . . . . . 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 805 . . . . . . . . . . 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 23995 . . . . . . . . . . 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 10071 . . . . . . . . . . 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 4372 . . . . . . . . . 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 3822 . . . . . . . . 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 10055 . . . . . . . . 9 ℂ ∈ V
11 ssexg 4837 . . . . . . . . 9 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ℂ ∈ V) → (𝑆 ∪ {0}) ∈ V)
129, 10, 11sylancl 695 . . . . . . . 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 11336 . . . . . . . 8 0 ∈ V
14 elmapg 7912 . . . . . . . 8 (((𝑆 ∪ {0}) ∈ V ∧ ℕ0 ∈ V) → (𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0) ↔ 𝑎:ℕ0⟶(𝑆 ∪ {0})))
1512, 13, 14sylancl 695 . . . . . . 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 222 . . . . . 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 817 . . . . . . . . 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 6095 . . . . . . . . 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 809 . . . . . . . . 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 811 . . . . . . . . 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 24028 . . . . . . . 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 2698 . . . . . . 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 6068 . . . . . 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 222 . . . . 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 449 . . . 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 3067 . . 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 11436 . . 3 0 ⊆ ℤ
30 ffn 6083 . . . . . . . . . . . . . . 15 (𝑎:ℕ0⟶ℂ → 𝑎 Fn ℕ0)
31 elpreima 6377 . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . 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 478 . . . . . . . . . . . 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 4353 . . . . . . . . . . . 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 474 . . . . . . . . . . . 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 23993 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0𝑎:ℕ0⟶ℂ) → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ ∀𝑥 ∈ ℕ0 ((𝑎𝑥) ≠ 0 → 𝑥𝑛)))
3918, 19, 38syl2anc 694 . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 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 2961 . . . . . . . . . . . 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 486 . . . . . . . . . . 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 2995 . . . . . . . . 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 5330 . . . . . . . . . . 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 5500 . . . . . . . . . 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 3174 . . . . . . . . 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 222 . . . . . . . 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 449 . . . . . . 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 642 . . . . . 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 3059 . . . . 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 3046 . . . 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 3700 . . 3 (ℕ0 ⊆ ℤ → (∃𝑛 ∈ ℕ0𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛 → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
5529, 53, 54mpsyl 68 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛)
5628, 55jca 553 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 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  wss 3607  {csn 4210   class class class wbr 4685  cmpt 4762  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  cc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  cexp 12900  Σcsu 14460  Polycply 23985  coeffccoe 23987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-0p 23482  df-ply 23989  df-coe 23991
This theorem is referenced by:  coef  24031  dgrcl  24034  dgrub  24035  dgrlb  24037
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