Theorem plyadd 26176

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 26155	. . . 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 496	. . 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 26155	. . . 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 496	. . 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 3206	. . 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 3206	. . . . 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 1198	. . . . . . . . 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 580	. . . . . . . 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 1239	. . . . . . . 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 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)))))) → 𝑛 ∈ ℕ0)
18		simp2l 1200	. . . . . . . 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 1201	. . . . . . . 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 1245	. . . . . . . 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 1247	. . . . . . . 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 1246	. . . . . . . . 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 7363	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
24	23	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
25	24	sumeq2sdv 15624	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)))
26		fveq2 6832	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
27		oveq2 7364	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
28	26, 27	oveq12d 7374	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
29	28	cbvsumv 15617	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗))
30	25, 29	eqtrdi 2785	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvmptv 5200	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
32	22, 31	eqtrdi 2785	. . . . . . . 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 1248	. . . . . . . . 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 7372	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
35	34	sumeq2sdv 15624	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)))
36		fveq2 6832	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
37	36, 27	oveq12d 7374	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
38	37	cbvsumv 15617	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗))
39	35, 38	eqtrdi 2785	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvmptv 5200	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
41	33, 40	eqtrdi 2785	. . . . . . . 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 26174	. . . . . . 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 1121	. . . . . 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 3191	. . . . 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	biimtrrid 243	. . . 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 3191	. . 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	biimtrrid 243	. 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 699	1 ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3058   ∪ cun 3897   ⊆ wss 3899  {csn 4578   ↦ cmpt 5177   “ cima 5625  ‘cfv 6490  (class class class)co 7356   ∘_f cof 7618   ↑_m cmap 8761  ℂcc 11022  0cc0 11024  1c1 11025   + caddc 11027   · cmul 11029  ℕ₀cn0 12399  ℤ_≥cuz 12749  ...cfz 13421  ↑cexp 13982  Σcsu 15607  Polycply 26143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-fzo 13569  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-sum 15608  df-ply 26147

This theorem is referenced by:  plysub  26178  plyaddcl  26179  plyco  26200  plydivlem4  26258  iaa  26287  rngunsnply  43353