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Theorem coeeu 26454

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 26428	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3927	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3		elply2 26424	. . . . . 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 503	. . . . 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 3291	. . . . 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 11225	. . . . . . 7 ⊢ 0 ∈ ℂ
9		snssi 4746	. . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ {0} ⊆ ℂ
11		ssequn2 4135	. . . . . 6 ⊢ ({0} ⊆ ℂ ↔ (ℂ ∪ {0}) = ℂ)
12	10, 11	mpbi 233	. . . . 5 ⊢ (ℂ ∪ {0}) = ℂ
13	12	oveq1i 7424	. . . 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 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 3234	. . . 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 7421	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
26	25	oveq2d 7430	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
27	26	sumeq2sdv 15793	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)))
28		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
29		oveq2 7422	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
30	28, 29	oveq12d 7432	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvsumv 15786	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗))
32	27, 31	eqtrdi 2811	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
33	32	cbvmptv 5209	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑎‘𝑗) · (𝑤↑𝑗)))
34	24, 33	eqtrdi 2811	. . . . . . 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 7430	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
37	36	sumeq2sdv 15793	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)))
38		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
39	38, 29	oveq12d 7432	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvsumv 15786	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗))
41	37, 40	eqtrdi 2811	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
42	41	cbvmptv 5209	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑏‘𝑗) · (𝑤↑𝑗)))
43	35, 42	eqtrdi 2811	. . . . . . 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 26453	. . . . . 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 3219	. . . 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 3205	. 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 6051	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑛 + 1))))
50	49	eqeq1d 2762	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑎 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0}))
51		fveq1 6878	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑘) = (𝑏‘𝑘))
52	51	oveq1d 7429	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑧↑𝑘)))
53	52	sumeq2sdv 15793	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))
54	53	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))
55	54	eqeq2d 2771	. . . . . 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 644	. . . . 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 3186	. . . 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 7437	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑚 + 1)))
59	58	imaeq2d 6056	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑏 “ (ℤ_≥‘(𝑛 + 1))) = (𝑏 “ (ℤ_≥‘(𝑚 + 1))))
60	59	eqeq1d 2762	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝑏 “ (ℤ_≥‘(𝑛 + 1))) = {0} ↔ (𝑏 “ (ℤ_≥‘(𝑚 + 1))) = {0}))
61		oveq2 7422	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
62	61	sumeq1d 15790	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘)))
63	62	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑏‘𝑘) · (𝑧↑𝑘))))
64	63	eqeq2d 2771	. . . . . 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 644	. . . . 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 3241	. . . 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 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 595	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 ∪ cun 3897 ⊆ wss 3899 {csn 4584 ↦ cmpt 5186 “ cima 5658 ‘cfv 6533 (class class class)co 7414 ↑_m cmap 8829 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ℕ₀cn0 12531 ℤ_≥cuz 12890 ...cfz 13564 ↑cexp 14128 Σcsu 15776 Polycply 26412

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13046 df-fz 13565 df-fzo 13713 df-fl 13856 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-rlim 15579 df-sum 15777 df-0p 25901 df-ply 26416

This theorem is used by: coelem 26455 coeeq 26456

W3C validator