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Theorem coeeu 26265

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 26240	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3932	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3		elply2 26236	. . . . . 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 501	. . . . 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 3290	. . . . 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 220	. . . 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 11168	. . . . . . 7 ⊢ 0 ∈ ℂ
9		snssi 4743	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
11		ssequn2 4141	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
12	10, 11	mpbi 232	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
13	12	oveq1i 7402	. . . 4 ⊢ ((ℂ ∪ {0}) ↑_m ℕ₀) = (ℂ ↑_m ℕ₀)
14	13	rexeqi 3318	. . 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 220	. 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 3233	. . . 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 1210	. . . . . . 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 1251	. . . . . . 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 1252	. . . . . . 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 1212	. . . . . . 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 1213	. . . . . . 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 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 “ (ℤ_≥‘(𝑛 + 1))) = {0})
23		simp3rl 1259	. . . . . . 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 1258	. . . . . . . 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 7399	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
26	25	oveq2d 7408	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
27	26	sumeq2sdv 15713	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)))
28		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
29		oveq2 7400	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
30	28, 29	oveq12d 7410	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvsumv 15706	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))
32	27, 31	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
33	32	cbvmptv 5203	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
34	24, 33	eqtrdi 2812	. . . . . . 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 1260	. . . . . . . 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 7408	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
37	36	sumeq2sdv 15713	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)))
38		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
39	38, 29	oveq12d 7410	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvsumv 15706	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗))
41	37, 40	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
42	41	cbvmptv 5203	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
43	35, 42	eqtrdi 2812	. . . . . . 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 26264	. . . . . 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 1133	. . . . 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 3218	. . . 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 245	. . 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 3204	. 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 6041	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑛 + 1))))
50	49	eqeq1d 2763	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0}))
51		fveq1 6862	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑘) = (𝑏‘𝑘))
52	51	oveq1d 7407	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑧↑𝑘)))
53	52	sumeq2sdv 15713	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))
54	53	mpteq2dv 5193	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))
55	54	eqeq2d 2772	. . . . . 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 641	. . . . 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 3185	. . . 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 7415	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑚 + 1)))
59	58	imaeq2d 6046	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑚 + 1))))
60	59	eqeq1d 2763	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0}))
61		oveq2 7400	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
62	61	sumeq1d 15710	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))
63	62	mpteq2dv 5193	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))
64	63	eqeq2d 2772	. . . . . 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 641	. . . . 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 3240	. . . 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 289	. . 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 3693	. 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 592	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 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ∃!wreu 3364 ∪ cun 3902 ⊆ wss 3904 {csn 4581 ↦ cmpt 5180 “ cima 5648 ‘cfv 6517 (class class class)co 7392 ↑_m cmap 8803 ℂcc 11068 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 ℕ₀cn0 12478 ℤ_≥cuz 12836 ...cfz 13509 ↑cexp 14071 Σcsu 15696 Polycply 26224

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-pm 8806 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-rp 12991 df-fz 13510 df-fzo 13657 df-fl 13799 df-seq 14012 df-exp 14072 df-hash 14341 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-clim 15498 df-rlim 15499 df-sum 15697 df-0p 25712 df-ply 26228

This theorem is referenced by: coelem 26266 coeeq 26267

W3C validator