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Theorem coeeu 25586
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 25561 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3940 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3 elply2 25557 . . . . . 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 497 . . . . 5 (𝐹 ∈ (Poly‘ℂ) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((ℂ ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
5 rexcom 3273 . . . . 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 217 . . . 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 11147 . . . . . . 7 0 ∈ ℂ
9 snssi 4768 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
108, 9ax-mp 5 . . . . . 6 {0} ⊆ ℂ
11 ssequn2 4143 . . . . . 6 ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
1210, 11mpbi 229 . . . . 5 (ℂ ∪ {0}) = ℂ
1312oveq1i 7367 . . . 4 ((ℂ ∪ {0}) ↑m0) = (ℂ ↑m0)
1413rexeqi 3312 . . 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 217 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 reeanv 3217 . . . 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 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)))))) → 𝐹 ∈ (Poly‘𝑆))
18 simp1rl 1238 . . . . . . 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 1239 . . . . . . 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 1199 . . . . . . 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 1200 . . . . . . 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 1244 . . . . . . 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 1246 . . . . . . 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 1245 . . . . . . . 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 7364 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑘) = (𝑤𝑘))
2625oveq2d 7373 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑎𝑘) · (𝑤𝑘)))
2726sumeq2sdv 15589 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
28 fveq2 6842 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑎𝑘) = (𝑎𝑗))
29 oveq2 7365 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
3028, 29oveq12d 7375 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑎𝑘) · (𝑤𝑘)) = ((𝑎𝑗) · (𝑤𝑗)))
3130cbvsumv 15581 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))
3227, 31eqtrdi 2792 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3332cbvmptv 5218 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3424, 33eqtrdi 2792 . . . . . . 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 1247 . . . . . . . 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 7373 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑏𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑤𝑘)))
3736sumeq2sdv 15589 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)))
38 fveq2 6842 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑏𝑘) = (𝑏𝑗))
3938, 29oveq12d 7375 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑏𝑘) · (𝑤𝑘)) = ((𝑏𝑗) · (𝑤𝑗)))
4039cbvsumv 15581 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗))
4137, 40eqtrdi 2792 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4241cbvmptv 5218 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4335, 42eqtrdi 2792 . . . . . . 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 25585 . . . . . 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 1121 . . . . 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 3205 . . . 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, 46biimtrrid 242 . . 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 3197 . 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 6008 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑛 + 1))))
5049eqeq1d 2738 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑛 + 1))) = {0}))
51 fveq1 6841 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎𝑘) = (𝑏𝑘))
5251oveq1d 7372 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑧𝑘)))
5352sumeq2sdv 15589 . . . . . . . 8 (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))
5453mpteq2dv 5207 . . . . . . 7 (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))
5554eqeq2d 2747 . . . . . 6 (𝑎 = 𝑏 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))))
5650, 55anbi12d 631 . . . . 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 3175 . . . 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 7380 . . . . . . . 8 (𝑛 = 𝑚 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑚 + 1)))
5958imaeq2d 6013 . . . . . . 7 (𝑛 = 𝑚 → (𝑏 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑚 + 1))))
6059eqeq1d 2738 . . . . . 6 (𝑛 = 𝑚 → ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑚 + 1))) = {0}))
61 oveq2 7365 . . . . . . . . 9 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
6261sumeq1d 15586 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))
6362mpteq2dv 5207 . . . . . . 7 (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))
6463eqeq2d 2747 . . . . . 6 (𝑛 = 𝑚 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))
6560, 64anbi12d 631 . . . . 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 3226 . . . 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, 66bitrdi 286 . . 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 3689 . 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 583 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351  cun 3908  wss 3910  {csn 4586  cmpt 5188  cima 5636  cfv 6496  (class class class)co 7357  m cmap 8765  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  0cn0 12413  cuz 12763  ...cfz 13424  cexp 13967  Σcsu 15570  Polycply 25545
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549
This theorem is referenced by:  coelem  25587  coeeq  25588
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