Theorem coeeu 26363

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 26338	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3934	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3		elply2 26334	. . . . . 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 502	. . . . 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 3294	. . . . 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 221	. . . 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 18	. . 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 11199	. . . . . . 7 ⊢ 0 ∈ ℂ
9		snssi 4752	. . . . . . 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 233	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
13	12	oveq1i 7422	. . . 4 ⊢ ((ℂ ∪ {0}) ↑m ℕ0) = (ℂ ↑m ℕ0)
14	13	rexeqi 3322	. . 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 221	. 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 3237	. . . 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 1216	. . . . . . 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 1257	. . . . . . 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 1258	. . . . . . 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 1218	. . . . . . 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 1219	. . . . . . 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 1263	. . . . . . 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 1265	. . . . . . 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 1264	. . . . . . . 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 7419	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
26	25	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
27	26	sumeq2sdv 15756	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)))
28		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
29		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
30	28, 29	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvsumv 15749	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))
32	27, 31	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
33	32	cbvmptv 5216	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
34	24, 33	eqtrdi 2814	. . . . . . 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 1266	. . . . . . . 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 7428	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
37	36	sumeq2sdv 15756	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)))
38		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
39	38, 29	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvsumv 15749	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗))
41	37, 40	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
42	41	cbvmptv 5216	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
43	35, 42	eqtrdi 2814	. . . . . . 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 26362	. . . . . 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 1139	. . . . 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 3222	. . . 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 246	. . 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 3208	. 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 6059	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 “ (ℤ≥‘(𝑛 + 1))) = (𝑏 “ (ℤ≥‘(𝑛 + 1))))
50	49	eqeq1d 2765	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0}))
51		fveq1 6882	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑘) = (𝑏‘𝑘))
52	51	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑧↑𝑘)))
53	52	sumeq2sdv 15756	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))
54	53	mpteq2dv 5206	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))
55	54	eqeq2d 2774	. . . . . 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 643	. . . . 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 3189	. . . 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 7435	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑚 + 1)))
59	58	imaeq2d 6064	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑏 “ (ℤ≥‘(𝑛 + 1))) = (𝑏 “ (ℤ≥‘(𝑚 + 1))))
60	59	eqeq1d 2765	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ≥‘(𝑚 + 1))) = {0}))
61		oveq2 7420	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
62	61	sumeq1d 15753	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))
63	62	mpteq2dv 5206	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))
64	63	eqeq2d 2774	. . . . . 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 643	. . . . 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 3244	. . . 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 290	. . 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 3695	. 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 594	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)))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367   ∪ cun 3904   ⊆ wss 3906  {csn 4590   ↦ cmpt 5193   “ cima 5666  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825  ℂcc 11099  0cc0 11101  1c1 11102   + caddc 11104   · cmul 11106  ℕ₀cn0 12505  ℤ_≥cuz 12863  ...cfz 13536  ↑cexp 14099  Σcsu 15739  Polycply 26322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-z 12593  df-uz 12864  df-rp 13018  df-fz 13537  df-fzo 13685  df-fl 13827  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-clim 15541  df-rlim 15542  df-sum 15740  df-0p 25810  df-ply 26326

This theorem is referenced by:  coelem  26364  coeeq  26365