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Theorem coeeu 23785
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 23760 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3564 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3 elply2 23756 . . . . . 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 479 . . . . 5 (𝐹 ∈ (Poly‘ℂ) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((ℂ ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
5 rexcom 3080 . . . . 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 207 . . . 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 9911 . . . . . . 7 0 ∈ ℂ
9 snssi 4280 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
108, 9ax-mp 5 . . . . . 6 {0} ⊆ ℂ
11 ssequn2 3748 . . . . . 6 ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
1210, 11mpbi 219 . . . . 5 (ℂ ∪ {0}) = ℂ
1312oveq1i 6559 . . . 4 ((ℂ ∪ {0}) ↑𝑚0) = (ℂ ↑𝑚0)
1413rexeqi 3120 . . 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 207 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 reeanv 3086 . . . 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 1078 . . . . . . 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 1119 . . . . . . 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 1120 . . . . . . 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 1080 . . . . . . 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 1081 . . . . . . 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 1125 . . . . . . 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 1127 . . . . . . 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 1126 . . . . . . . 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 6556 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑘) = (𝑤𝑘))
2625oveq2d 6565 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑎𝑘) · (𝑤𝑘)))
2726sumeq2sdv 14282 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
28 fveq2 6103 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑎𝑘) = (𝑎𝑗))
29 oveq2 6557 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
3028, 29oveq12d 6567 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑎𝑘) · (𝑤𝑘)) = ((𝑎𝑗) · (𝑤𝑗)))
3130cbvsumv 14274 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))
3227, 31syl6eq 2660 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3332cbvmptv 4678 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3424, 33syl6eq 2660 . . . . . . 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 1128 . . . . . . . 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 6565 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑏𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑤𝑘)))
3736sumeq2sdv 14282 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)))
38 fveq2 6103 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑏𝑘) = (𝑏𝑗))
3938, 29oveq12d 6567 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑏𝑘) · (𝑤𝑘)) = ((𝑏𝑗) · (𝑤𝑗)))
4039cbvsumv 14274 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗))
4137, 40syl6eq 2660 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4241cbvmptv 4678 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4335, 42syl6eq 2660 . . . . . . 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 23784 . . . . . 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 1259 . . . . 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 3020 . . . 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 232 . . 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 2954 . 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 5380 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑛 + 1))))
5049eqeq1d 2612 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑛 + 1))) = {0}))
51 fveq1 6102 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎𝑘) = (𝑏𝑘))
5251oveq1d 6564 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑧𝑘)))
5352sumeq2sdv 14282 . . . . . . . 8 (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))
5453mpteq2dv 4673 . . . . . . 7 (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))
5554eqeq2d 2620 . . . . . 6 (𝑎 = 𝑏 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))))
5650, 55anbi12d 743 . . . . 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 3034 . . . 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 oveq1 6556 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
5958fveq2d 6107 . . . . . . . 8 (𝑛 = 𝑚 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑚 + 1)))
6059imaeq2d 5385 . . . . . . 7 (𝑛 = 𝑚 → (𝑏 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑚 + 1))))
6160eqeq1d 2612 . . . . . 6 (𝑛 = 𝑚 → ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑚 + 1))) = {0}))
62 oveq2 6557 . . . . . . . . 9 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
6362sumeq1d 14279 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))
6463mpteq2dv 4673 . . . . . . 7 (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))
6564eqeq2d 2620 . . . . . 6 (𝑛 = 𝑚 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))
6661, 65anbi12d 743 . . . . 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))))))
6766cbvrexv 3148 . . . 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)))))
6857, 67syl6bb 275 . . 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))))))
6968reu4 3367 . 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)))
7015, 48, 69sylanbrc 695 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 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  ∃!wreu 2898  cun 3538  wss 3540  {csn 4125  cmpt 4643  cima 5041  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  cc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  0cn0 11169  cuz 11563  ...cfz 12197  cexp 12722  Σcsu 14264  Polycply 23744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748
This theorem is referenced by:  coelem  23786  coeeq  23787
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