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Theorem coeeu 24825
Description: Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
coeeu (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
Distinct variable groups:   𝑧,𝑘   𝑛,𝑎,𝐹   𝑆,𝑎,𝑛   𝑘,𝑎,𝑧,𝑛
Allowed substitution hints:   𝑆(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem coeeu
Dummy variables 𝑏 𝑗 𝑚 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 24800 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3949 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3 elply2 24796 . . . . . 6 (𝐹 ∈ (Poly‘ℂ) ↔ (ℂ ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((ℂ ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
43simprbi 500 . . . . 5 (𝐹 ∈ (Poly‘ℂ) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((ℂ ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
5 rexcom 3346 . . . . 5 (∃𝑛 ∈ ℕ0𝑎 ∈ ((ℂ ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ∃𝑎 ∈ ((ℂ ∪ {0}) ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
64, 5sylib 221 . . . 4 (𝐹 ∈ (Poly‘ℂ) → ∃𝑎 ∈ ((ℂ ∪ {0}) ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
72, 6syl 17 . . 3 (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ ((ℂ ∪ {0}) ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
8 0cn 10631 . . . . . . 7 0 ∈ ℂ
9 snssi 4725 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
108, 9ax-mp 5 . . . . . 6 {0} ⊆ ℂ
11 ssequn2 4145 . . . . . 6 ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
1210, 11mpbi 233 . . . . 5 (ℂ ∪ {0}) = ℂ
1312oveq1i 7159 . . . 4 ((ℂ ∪ {0}) ↑m0) = (ℂ ↑m0)
1413rexeqi 3401 . . 3 (∃𝑎 ∈ ((ℂ ∪ {0}) ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ∃𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
157, 14sylib 221 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 reeanv 3358 . . . 4 (∃𝑛 ∈ ℕ0𝑚 ∈ ℕ0 (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))) ↔ (∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ∃𝑚 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))))
17 simp1l 1194 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝐹 ∈ (Poly‘𝑆))
18 simp1rl 1235 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝑎 ∈ (ℂ ↑m0))
19 simp1rr 1236 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝑏 ∈ (ℂ ↑m0))
20 simp2l 1196 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝑛 ∈ ℕ0)
21 simp2r 1197 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝑚 ∈ ℕ0)
22 simp3ll 1241 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → (𝑎 “ (ℤ‘(𝑛 + 1))) = {0})
23 simp3rl 1243 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → (𝑏 “ (ℤ‘(𝑚 + 1))) = {0})
24 simp3lr 1242 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))
25 oveq1 7156 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑘) = (𝑤𝑘))
2625oveq2d 7165 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑎𝑘) · (𝑤𝑘)))
2726sumeq2sdv 15061 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
28 fveq2 6661 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑎𝑘) = (𝑎𝑗))
29 oveq2 7157 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
3028, 29oveq12d 7167 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑎𝑘) · (𝑤𝑘)) = ((𝑎𝑗) · (𝑤𝑗)))
3130cbvsumv 15053 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))
3227, 31syl6eq 2875 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3332cbvmptv 5155 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3424, 33syl6eq 2875 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))))
35 simp3rr 1244 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))
3625oveq2d 7165 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑏𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑤𝑘)))
3736sumeq2sdv 15061 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)))
38 fveq2 6661 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑏𝑘) = (𝑏𝑗))
3938, 29oveq12d 7167 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑏𝑘) · (𝑤𝑘)) = ((𝑏𝑗) · (𝑤𝑗)))
4039cbvsumv 15053 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗))
4137, 40syl6eq 2875 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4241cbvmptv 5155 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4335, 42syl6eq 2875 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗))))
4417, 18, 19, 20, 21, 22, 23, 34, 43coeeulem 24824 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0) ∧ (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))) → 𝑎 = 𝑏)
45443expia 1118 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) ∧ (𝑛 ∈ ℕ0𝑚 ∈ ℕ0)) → ((((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))) → 𝑎 = 𝑏))
4645rexlimdvva 3286 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) → (∃𝑛 ∈ ℕ0𝑚 ∈ ℕ0 (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))) → 𝑎 = 𝑏))
4716, 46syl5bir 246 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑m0) ∧ 𝑏 ∈ (ℂ ↑m0))) → ((∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ∃𝑚 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))) → 𝑎 = 𝑏))
4847ralrimivva 3186 . 2 (𝐹 ∈ (Poly‘𝑆) → ∀𝑎 ∈ (ℂ ↑m0)∀𝑏 ∈ (ℂ ↑m0)((∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ∃𝑚 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))) → 𝑎 = 𝑏))
49 imaeq1 5911 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑛 + 1))))
5049eqeq1d 2826 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑛 + 1))) = {0}))
51 fveq1 6660 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎𝑘) = (𝑏𝑘))
5251oveq1d 7164 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑧𝑘)))
5352sumeq2sdv 15061 . . . . . . . 8 (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))
5453mpteq2dv 5148 . . . . . . 7 (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))
5554eqeq2d 2835 . . . . . 6 (𝑎 = 𝑏 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))))
5650, 55anbi12d 633 . . . . 5 (𝑎 = 𝑏 → (((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))))
5756rexbidv 3289 . . . 4 (𝑎 = 𝑏 → (∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ∃𝑛 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))))
58 fvoveq1 7172 . . . . . . . 8 (𝑛 = 𝑚 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑚 + 1)))
5958imaeq2d 5916 . . . . . . 7 (𝑛 = 𝑚 → (𝑏 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑚 + 1))))
6059eqeq1d 2826 . . . . . 6 (𝑛 = 𝑚 → ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑚 + 1))) = {0}))
61 oveq2 7157 . . . . . . . . 9 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
6261sumeq1d 15058 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))
6362mpteq2dv 5148 . . . . . . 7 (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))
6463eqeq2d 2835 . . . . . 6 (𝑛 = 𝑚 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))
6560, 64anbi12d 633 . . . . 5 (𝑛 = 𝑚 → (((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))) ↔ ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))))
6665cbvrexvw 3435 . . . 4 (∃𝑛 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))) ↔ ∃𝑚 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))
6757, 66syl6bb 290 . . 3 (𝑎 = 𝑏 → (∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ ∃𝑚 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))))
6867reu4 3708 . 2 (∃!𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ↔ (∃𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ∀𝑎 ∈ (ℂ ↑m0)∀𝑏 ∈ (ℂ ↑m0)((∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))) ∧ ∃𝑚 ∈ ℕ0 ((𝑏 “ (ℤ‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))) → 𝑎 = 𝑏)))
6915, 48, 68sylanbrc 586 1 (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  ∃!wreu 3135  cun 3917  wss 3919  {csn 4550  cmpt 5132  cima 5545  cfv 6343  (class class class)co 7149  m cmap 8402  cc 10533  0cc0 10535  1c1 10536   + caddc 10538   · cmul 10540  0cn0 11894  cuz 12240  ...cfz 12894  cexp 13434  Σcsu 15042  Polycply 24784
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-0p 24277  df-ply 24788
This theorem is referenced by:  coelem  24826  coeeq  24827
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