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Theorem coeeu 26264
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 26239 . . . . 5 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3979 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3 elply2 26235 . . . . . 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 496 . . . . 5 (𝐹 ∈ (Poly‘ℂ) → ∃𝑛 ∈ ℕ0𝑎 ∈ ((ℂ ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
5 rexcom 3290 . . . . 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 218 . . . 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 11253 . . . . . . 7 0 ∈ ℂ
9 snssi 4808 . . . . . . 7 (0 ∈ ℂ → {0} ⊆ ℂ)
108, 9ax-mp 5 . . . . . 6 {0} ⊆ ℂ
11 ssequn2 4189 . . . . . 6 ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
1210, 11mpbi 230 . . . . 5 (ℂ ∪ {0}) = ℂ
1312oveq1i 7441 . . . 4 ((ℂ ∪ {0}) ↑m0) = (ℂ ↑m0)
1413rexeqi 3325 . . 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 218 . 2 (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
16 reeanv 3229 . . . 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 1198 . . . . . . 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 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)))))) → 𝑎 ∈ (ℂ ↑m0))
19 simp1rr 1240 . . . . . . 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 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)))))) → 𝑛 ∈ ℕ0)
21 simp2r 1201 . . . . . . 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 1245 . . . . . . 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 1247 . . . . . . 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 1246 . . . . . . . 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 7438 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑘) = (𝑤𝑘))
2625oveq2d 7447 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑎𝑘) · (𝑤𝑘)))
2726sumeq2sdv 15739 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)))
28 fveq2 6906 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑎𝑘) = (𝑎𝑗))
29 oveq2 7439 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑤𝑘) = (𝑤𝑗))
3028, 29oveq12d 7449 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑎𝑘) · (𝑤𝑘)) = ((𝑎𝑗) · (𝑤𝑗)))
3130cbvsumv 15732 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗))
3227, 31eqtrdi 2793 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3332cbvmptv 5255 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎𝑗) · (𝑤𝑗)))
3424, 33eqtrdi 2793 . . . . . . 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 1248 . . . . . . . 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 7447 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝑏𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑤𝑘)))
3736sumeq2sdv 15739 . . . . . . . . . 10 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)))
38 fveq2 6906 . . . . . . . . . . . 12 (𝑘 = 𝑗 → (𝑏𝑘) = (𝑏𝑗))
3938, 29oveq12d 7449 . . . . . . . . . . 11 (𝑘 = 𝑗 → ((𝑏𝑘) · (𝑤𝑘)) = ((𝑏𝑗) · (𝑤𝑗)))
4039cbvsumv 15732 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑤𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗))
4137, 40eqtrdi 2793 . . . . . . . . 9 (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4241cbvmptv 5255 . . . . . . . 8 (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏𝑗) · (𝑤𝑗)))
4335, 42eqtrdi 2793 . . . . . . 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 26263 . . . . . 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 1122 . . . . 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 3213 . . . 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 243 . . 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 3202 . 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 6073 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑛 + 1))))
5049eqeq1d 2739 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑛 + 1))) = {0}))
51 fveq1 6905 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎𝑘) = (𝑏𝑘))
5251oveq1d 7446 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝑎𝑘) · (𝑧𝑘)) = ((𝑏𝑘) · (𝑧𝑘)))
5352sumeq2sdv 15739 . . . . . . . 8 (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))
5453mpteq2dv 5244 . . . . . . 7 (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))))
5554eqeq2d 2748 . . . . . 6 (𝑎 = 𝑏 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)))))
5650, 55anbi12d 632 . . . . 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 3179 . . . 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 7454 . . . . . . . 8 (𝑛 = 𝑚 → (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑚 + 1)))
5958imaeq2d 6078 . . . . . . 7 (𝑛 = 𝑚 → (𝑏 “ (ℤ‘(𝑛 + 1))) = (𝑏 “ (ℤ‘(𝑚 + 1))))
6059eqeq1d 2739 . . . . . 6 (𝑛 = 𝑚 → ((𝑏 “ (ℤ‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ‘(𝑚 + 1))) = {0}))
61 oveq2 7439 . . . . . . . . 9 (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
6261sumeq1d 15736 . . . . . . . 8 (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))
6362mpteq2dv 5244 . . . . . . 7 (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘))))
6463eqeq2d 2748 . . . . . 6 (𝑛 = 𝑚 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏𝑘) · (𝑧𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏𝑘) · (𝑧𝑘)))))
6560, 64anbi12d 632 . . . . 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 3238 . . . 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 287 . . 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 3737 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378  cun 3949  wss 3951  {csn 4626  cmpt 5225  cima 5688  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  0cn0 12526  cuz 12878  ...cfz 13547  cexp 14102  Σcsu 15722  Polycply 26223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227
This theorem is referenced by:  coelem  26265  coeeq  26266
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