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Theorem plymul 26145

Description: The product 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)
plymul.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)

**Assertion**
Ref	Expression
plymul	⊢ (𝜑 → (𝐹 ∘_f · 𝐺) ∈ (Poly‘𝑆))

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

Proof of Theorem plymul 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 26123	. . . 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 26123	. . . 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 3221	. . 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 3221	. . . . 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 7421	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
24	23	oveq2d 7430	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
25	24	sumeq2sdv 15676	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)))
26		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
27		oveq2 7422	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
28	26, 27	oveq12d 7432	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
29	28	cbvsumv 15668	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗))
30	25, 29	eqtrdi 2783	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvmptv 5255	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
32	22, 31	eqtrdi 2783	. . . . . . . 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 7430	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
35	34	sumeq2sdv 15676	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)))
36		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
37	36, 27	oveq12d 7432	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
38	37	cbvsumv 15668	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗))
39	35, 38	eqtrdi 2783	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvmptv 5255	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
41	33, 40	eqtrdi 2783	. . . . . . . 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		plymul.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
43	11, 42	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)
44	12, 13, 15, 16, 17, 18, 19, 20, 21, 32, 41, 43	plymullem 26143	. . . . . . 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))
45	44	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)))
46	45	rexlimdvva 3206	. . . . 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)))
47	10, 46	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)))
48	47	rexlimdvva 3206	. . 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)))
49	9, 48	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)))
50	4, 8, 49	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 3065 ∪ cun 3942 ⊆ wss 3944 {csn 4624 ↦ cmpt 5225 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ∘_f cof 7677 ↑_m cmap 8838 ℂcc 11130 0cc0 11132 1c1 11133 + caddc 11135 · cmul 11137 ℕ₀cn0 12496 ℤ_≥cuz 12846 ...cfz 13510 ↑cexp 14052 Σcsu 15658 Polycply 26111

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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-ply 26115

This theorem is referenced by: plysub 26146 plymulcl 26148 plyco 26168 plydivlem2 26222 plydivlem4 26224 plydiveu 26226 plymulx0 34169 mpaaeu 42546 rngunsnply 42569

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