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Theorem coeeu 24026
 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 24001 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3632 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3 elply2 23997 . . . . . 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 3128 . . . . 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 208 . . . 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 10070 . . . . . . 7 0 ∈ ℂ
9 snssi 4371 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
108, 9ax-mp 5 . . . . . 6 {0} ⊆ ℂ
11 ssequn2 3819 . . . . . 6 ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
1210, 11mpbi 220 . . . . 5 (ℂ ∪ {0}) = ℂ
1312oveq1i 6700 . . . 4 ((ℂ ∪ {0}) ↑𝑚0) = (ℂ ↑𝑚0)
1413rexeqi 3173 . . 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 208 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 reeanv 3136 . . . 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 1105 . . . . . . 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 1146 . . . . . . 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 1147 . . . . . . 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 1107 . . . . . . 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 1108 . . . . . . 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 1152 . . . . . . 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 1154 . . . . . . 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 1153 . . . . . . . 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 6697 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑘) = (𝑤𝑘))
2625oveq2d 6706 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑎𝑘) · (𝑤𝑘)))
2726sumeq2sdv 14479 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
28 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑎𝑘) = (𝑎𝑗))
29 oveq2 6698 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
3028, 29oveq12d 6708 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑎𝑘) · (𝑤𝑘)) = ((𝑎𝑗) · (𝑤𝑗)))
3130cbvsumv 14470 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))
3227, 31syl6eq 2701 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3332cbvmptv 4783 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3424, 33syl6eq 2701 . . . . . . 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 1155 . . . . . . . 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 6706 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑏𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑤𝑘)))
3736sumeq2sdv 14479 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)))
38 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑏𝑘) = (𝑏𝑗))
3938, 29oveq12d 6708 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑏𝑘) · (𝑤𝑘)) = ((𝑏𝑗) · (𝑤𝑗)))
4039cbvsumv 14470 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗))
4137, 40syl6eq 2701 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4241cbvmptv 4783 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4335, 42syl6eq 2701 . . . . . . 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 24025 . . . . . 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 1286 . . . . 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 3067 . . . 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 233 . . 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 3000 . 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 5496 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑛 + 1))))
5049eqeq1d 2653 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑛 + 1))) = {0}))
51 fveq1 6228 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎𝑘) = (𝑏𝑘))
5251oveq1d 6705 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑧𝑘)))
5352sumeq2sdv 14479 . . . . . . . 8 (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))
5453mpteq2dv 4778 . . . . . . 7 (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))
5554eqeq2d 2661 . . . . . 6 (𝑎 = 𝑏 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))))
5650, 55anbi12d 747 . . . . 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 3081 . . . 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 6697 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
5958fveq2d 6233 . . . . . . . 8 (𝑛 = 𝑚 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑚 + 1)))
6059imaeq2d 5501 . . . . . . 7 (𝑛 = 𝑚 → (𝑏 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑚 + 1))))
6160eqeq1d 2653 . . . . . 6 (𝑛 = 𝑚 → ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑚 + 1))) = {0}))
62 oveq2 6698 . . . . . . . . 9 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
6362sumeq1d 14475 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))
6463mpteq2dv 4778 . . . . . . 7 (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))
6564eqeq2d 2661 . . . . . 6 (𝑛 = 𝑚 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))
6661, 65anbi12d 747 . . . . 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 3202 . . . 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 276 . . 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 3433 . 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 699 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 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943   ∪ cun 3605   ⊆ wss 3607  {csn 4210   ↦ cmpt 4762   “ cima 5146  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  ℂcc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  ℕ0cn0 11330  ℤ≥cuz 11725  ...cfz 12364  ↑cexp 12900  Σcsu 14460  Polycply 23985 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 This theorem is referenced by:  coelem  24027  coeeq  24028
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