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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 ↑_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 25561	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3940	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3		elply2 25557	. . . . . 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 497	. . . . 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 3273	. . . . 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 11147	. . . . . . 7 ⊢ 0 ∈ ℂ
9		snssi 4768	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
11		ssequn2 4143	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
12	10, 11	mpbi 229	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
13	12	oveq1i 7367	. . . 4 ⊢ ((ℂ ∪ {0}) ↑_m ℕ₀) = (ℂ ↑_m ℕ₀)
14	13	rexeqi 3312	. . 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 3217	. . . 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 1197	. . . . . . 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 1238	. . . . . . 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 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)))))) → 𝑏 ∈ (ℂ ↑_m ℕ₀))
20		simp2l 1199	. . . . . . 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 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)))))) → 𝑚 ∈ ℕ₀)
22		simp3ll 1244	. . . . . . 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 1246	. . . . . . 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 1245	. . . . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
26	25	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
27	26	sumeq2sdv 15589	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)))
28		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
29		oveq2 7365	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
30	28, 29	oveq12d 7375	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvsumv 15581	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))
32	27, 31	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
33	32	cbvmptv 5218	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
34	24, 33	eqtrdi 2792	. . . . . . 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 1247	. . . . . . . 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 7373	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
37	36	sumeq2sdv 15589	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)))
38		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
39	38, 29	oveq12d 7375	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvsumv 15581	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗))
41	37, 40	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
42	41	cbvmptv 5218	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
43	35, 42	eqtrdi 2792	. . . . . . 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 25585	. . . . . 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 1121	. . . . 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 3205	. . . 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 3197	. 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 6008	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑛 + 1))))
50	49	eqeq1d 2738	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0}))
51		fveq1 6841	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑘) = (𝑏‘𝑘))
52	51	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑧↑𝑘)))
53	52	sumeq2sdv 15589	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))
54	53	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))
55	54	eqeq2d 2747	. . . . . 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 631	. . . . 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 3175	. . . 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 7380	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑚 + 1)))
59	58	imaeq2d 6013	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑚 + 1))))
60	59	eqeq1d 2738	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0}))
61		oveq2 7365	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
62	61	sumeq1d 15586	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))
63	62	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))
64	63	eqeq2d 2747	. . . . . 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 631	. . . . 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 3226	. . . 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 286	. . 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 3689	. 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 583	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 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 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ℕ₀cn0 12413 ℤ_≥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

W3C validator