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Theorem plyadd 26164

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 26143	. . . 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 26143	. . . 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 3223	. . 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 3223	. . . . 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 1195	. . . . . . . . 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 579	. . . . . . . 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 1236	. . . . . . . 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 1237	. . . . . . . 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 1197	. . . . . . . 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 1198	. . . . . . . 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 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)))))) → (𝑎 “ (ℤ_≥‘(𝑚 + 1))) = {0})
21		simp3rl 1244	. . . . . . . 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 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)))))) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))))
23		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
24	23	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
25	24	sumeq2sdv 15683	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)))
26		fveq2 6897	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
27		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
28	26, 27	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
29	28	cbvsumv 15675	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗))
30	25, 29	eqtrdi 2784	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvmptv 5261	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
32	22, 31	eqtrdi 2784	. . . . . . . 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 1245	. . . . . . . . 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 7436	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
35	34	sumeq2sdv 15683	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)))
36		fveq2 6897	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
37	36, 27	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
38	37	cbvsumv 15675	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗))
39	35, 38	eqtrdi 2784	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvmptv 5261	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
41	33, 40	eqtrdi 2784	. . . . . . . 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 26162	. . . . . . 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 1119	. . . . . 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 3208	. . . . 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 242	. . . 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 3208	. . 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 242	. 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 698	1 ⊢ (𝜑 → (𝐹 ∘_f + 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 ∪ cun 3945 ⊆ wss 3947 {csn 4629 ↦ cmpt 5231 “ cima 5681 ‘cfv 6548 (class class class)co 7420 ∘_f cof 7683 ↑_m cmap 8845 ℂcc 11137 0cc0 11139 1c1 11140 + caddc 11142 · cmul 11144 ℕ₀cn0 12503 ℤ_≥cuz 12853 ...cfz 13517 ↑cexp 14059 Σcsu 15665 Polycply 26131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-ply 26135

This theorem is referenced by: plysub 26166 plyaddcl 26167 plyco 26188 plydivlem4 26244 iaa 26273 rngunsnply 42597

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