Theorem plyadd 24801

Description: The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)

Hypotheses

Ref	Expression
plyadd.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
plyadd.2	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
plyadd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
plyadd	⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑆,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦

Proof of Theorem plyadd

Dummy variables 𝑘 𝑚 𝑛 𝑧 𝑎 𝑏 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyadd.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
2		elply2 24780	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑚 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))))))
3	2	simprbi 499	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑚 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))))
4	1, 3	syl 17	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))))
5		plyadd.2	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
6		elply2 24780	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))))
7	6	simprbi 499	. . 3 ⊢ (𝐺 ∈ (Poly‘𝑆) → ∃𝑛 ∈ ℕ0 ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))
8	5, 7	syl 17	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))
9		reeanv 3367	. . 3 ⊢ (∃𝑚 ∈ ℕ0 ∃𝑛 ∈ ℕ0 (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) ↔ (∃𝑚 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑛 ∈ ℕ0 ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))))
10		reeanv 3367	. . . . 5 ⊢ (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)(((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) ↔ (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))))
11		simp1l 1193	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝜑)
12	11, 1	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 ∈ (Poly‘𝑆))
13	11, 5	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐺 ∈ (Poly‘𝑆))
14		plyadd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
15	11, 14	sylan 582	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
16		simp1rl 1234	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑚 ∈ ℕ0)
17		simp1rr 1235	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑛 ∈ ℕ0)
18		simp2l 1195	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))
19		simp2r 1196	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))
20		simp3ll 1240	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → (𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0})
21		simp3rl 1242	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → (𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0})
22		simp3lr 1241	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))))
23		oveq1 7157	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
24	23	oveq2d 7166	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
25	24	sumeq2sdv 15055	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)))
26		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
27		oveq2 7158	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
28	26, 27	oveq12d 7168	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
29	28	cbvsumv 15047	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗))
30	25, 29	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvmptv 5161	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
32	22, 31	syl6eq 2872	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐹 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗))))
33		simp3rr 1243	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))
34	23	oveq2d 7166	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
35	34	sumeq2sdv 15055	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)))
36		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
37	36, 27	oveq12d 7168	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
38	37	cbvsumv 15047	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗))
39	35, 38	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvmptv 5161	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
41	33, 40	syl6eq 2872	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → 𝐺 = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗))))
42	12, 13, 15, 16, 17, 18, 19, 20, 21, 32, 41	plyaddlem 24799	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) ∧ (((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)))))) → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆))
43	42	3expia 1117	. . . . . 6 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) ∧ (𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0) ∧ 𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))) → ((((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)))
44	43	rexlimdvva 3294	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)(((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)))
45	10, 44	syl5bir 245	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → ((∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)))
46	45	rexlimdvva 3294	. . 3 ⊢ (𝜑 → (∃𝑚 ∈ ℕ0 ∃𝑛 ∈ ℕ0 (∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)))
47	9, 46	syl5bir 245	. 2 ⊢ (𝜑 → ((∃𝑚 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑚 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)))) ∧ ∃𝑛 ∈ ℕ0 ∃𝑏 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑏 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))))) → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)))
48	4, 8, 47	mp2and 697	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139   ∪ cun 3933   ⊆ wss 3935  {csn 4560   ↦ cmpt 5138   “ cima 5552  ‘cfv 6349  (class class class)co 7150   ∘_f cof 7401   ↑_m cmap 8400  ℂcc 10529  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  ℕ₀cn0 11891  ℤ_≥cuz 12237  ...cfz 12886  ↑cexp 13423  Σcsu 15036  Polycply 24768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-fz 12887  df-fzo 13028  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-clim 14839  df-sum 15037  df-ply 24772

This theorem is referenced by:  plysub  24803  plyaddcl  24804  plyco  24825  plydivlem4  24879  iaa  24908  rngunsnply  39766