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Theorem coeeu 25739

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 ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))

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.

Step	Hyp	Ref	Expression
1		plyssc 25714	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3979	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3		elply2 25710	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℂ) ↔ (ℂ ⊆ ℂ ∧ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
4	3	simprbi 498	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℂ) → ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
5		rexcom 3288	. . . . 5 ⊢ (∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
6	4, 5	sylib 217	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℂ) → ∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
7	2, 6	syl 17	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
8		0cn 11206	. . . . . . 7 ⊢ 0 ∈ ℂ
9		snssi 4812	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
11		ssequn2 4184	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
12	10, 11	mpbi 229	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
13	12	oveq1i 7419	. . . 4 ⊢ ((ℂ ∪ {0}) ↑_m ℕ₀) = (ℂ ↑_m ℕ₀)
14	13	rexeqi 3325	. . 3 ⊢ (∃𝑎 ∈ ((ℂ ∪ {0}) ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ∃𝑎 ∈ (ℂ ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
15	7, 14	sylib 217	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑎 ∈ (ℂ ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
16		reeanv 3227	. . . 4 ⊢ (∃𝑛 ∈ ℕ₀ ∃𝑚 ∈ ℕ₀ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))) ↔ (∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑚 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))))
17		simp1l 1198	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 ∈ (Poly‘𝑆))
18		simp1rl 1239	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑎 ∈ (ℂ ↑_m ℕ₀))
19		simp1rr 1240	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑏 ∈ (ℂ ↑_m ℕ₀))
20		simp2l 1200	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑛 ∈ ℕ₀)
21		simp2r 1201	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑚 ∈ ℕ₀)
22		simp3ll 1245	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → (𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0})
23		simp3rl 1247	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → (𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0})
24		simp3lr 1246	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))
25		oveq1 7416	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
26	25	oveq2d 7425	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
27	26	sumeq2sdv 15650	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)))
28		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
29		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
30	28, 29	oveq12d 7427	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvsumv 15642	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))
32	27, 31	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
33	32	cbvmptv 5262	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
34	24, 33	eqtrdi 2789	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))))
35		simp3rr 1248	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))
36	25	oveq2d 7425	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
37	36	sumeq2sdv 15650	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)))
38		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
39	38, 29	oveq12d 7427	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvsumv 15642	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗))
41	37, 40	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
42	41	cbvmptv 5262	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
43	35, 42	eqtrdi 2789	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗))))
44	17, 18, 19, 20, 21, 22, 23, 34, 43	coeeulem 25738	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀) ∧ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑎 = 𝑏)
45	44	3expia 1122	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑚 ∈ ℕ₀)) → ((((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))) → 𝑎 = 𝑏))
46	45	rexlimdvva 3212	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) → (∃𝑛 ∈ ℕ₀ ∃𝑚 ∈ ℕ₀ (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))) → 𝑎 = 𝑏))
47	16, 46	biimtrrid 242	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ (𝑎 ∈ (ℂ ↑_m ℕ₀) ∧ 𝑏 ∈ (ℂ ↑_m ℕ₀))) → ((∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑚 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))) → 𝑎 = 𝑏))
48	47	ralrimivva 3201	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑎 ∈ (ℂ ↑_m ℕ₀)∀𝑏 ∈ (ℂ ↑_m ℕ₀)((∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑚 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))) → 𝑎 = 𝑏))
49		imaeq1 6055	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑛 + 1))))
50	49	eqeq1d 2735	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0}))
51		fveq1 6891	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑘) = (𝑏‘𝑘))
52	51	oveq1d 7424	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑧↑𝑘)))
53	52	sumeq2sdv 15650	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))
54	53	mpteq2dv 5251	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))
55	54	eqeq2d 2744	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))
56	50, 55	anbi12d 632	. . . . 5 ⊢ (𝑎 = 𝑏 → (((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))))
57	56	rexbidv 3179	. . . 4 ⊢ (𝑎 = 𝑏 → (∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ∃𝑛 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))))
58		fvoveq1 7432	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑚 + 1)))
59	58	imaeq2d 6060	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑚 + 1))))
60	59	eqeq1d 2735	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0}))
61		oveq2 7417	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
62	61	sumeq1d 15647	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))
63	62	mpteq2dv 5251	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))
64	63	eqeq2d 2744	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) ↔ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))
65	60, 64	anbi12d 632	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))) ↔ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))))
66	65	cbvrexvw 3236	. . . 4 ⊢ (∃𝑛 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))) ↔ ∃𝑚 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))))
67	57, 66	bitrdi 287	. . 3 ⊢ (𝑎 = 𝑏 → (∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ ∃𝑚 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))))
68	67	reu4 3728	. 2 ⊢ (∃!𝑎 ∈ (ℂ ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ (∃𝑎 ∈ (ℂ ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∀𝑎 ∈ (ℂ ↑_m ℕ₀)∀𝑏 ∈ (ℂ ↑_m ℕ₀)((∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑚 ∈ ℕ₀ ((𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))) → 𝑎 = 𝑏)))
69	15, 48, 68	sylanbrc 584	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑_m ℕ₀)∃𝑛 ∈ ℕ₀ ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 ∪ cun 3947 ⊆ wss 3949 {csn 4629 ↦ cmpt 5232 “ cima 5680 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ℕ₀cn0 12472 ℤ_≥cuz 12822 ...cfz 13484 ↑cexp 14027 Σcsu 15632 Polycply 25698

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-0p 25187 df-ply 25702

This theorem is referenced by: coelem 25740 coeeq 25741

W3C validator