Theorem plymul 26426

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 26404	. . . 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 503	. . 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 18	. 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 26404	. . . 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 503	. . 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 18	. 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 3239	. . 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 3239	. . . . 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 1216	. . . . . . . . 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 18	. . . . . . . 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 18	. . . . . . . 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 592	. . . . . . . 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 1257	. . . . . . . 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 1258	. . . . . . . 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 1218	. . . . . . . 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 1219	. . . . . . . 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 1263	. . . . . . . 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 1265	. . . . . . . 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 1264	. . . . . . . . 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 7426	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧↑𝑘) = (𝑤↑𝑘))
24	23	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑎‘𝑘) · (𝑧↑𝑘)) = ((𝑎‘𝑘) · (𝑤↑𝑘)))
25	24	sumeq2sdv 15778	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)))
26		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑎‘𝑘) = (𝑎‘𝑗))
27		oveq2 7427	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑤↑𝑘) = (𝑤↑𝑗))
28	26, 27	oveq12d 7437	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑎‘𝑘) · (𝑤↑𝑘)) = ((𝑎‘𝑗) · (𝑤↑𝑗)))
29	28	cbvsumv 15771	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗))
30	25, 29	eqtrdi 2816	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
31	30	cbvmptv 5217	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑚)((𝑎‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑚)((𝑎‘𝑗) · (𝑤↑𝑗)))
32	22, 31	eqtrdi 2816	. . . . . . . 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 1266	. . . . . . . . 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 7435	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝑏‘𝑘) · (𝑧↑𝑘)) = ((𝑏‘𝑘) · (𝑤↑𝑘)))
35	34	sumeq2sdv 15778	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)))
36		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑏‘𝑘) = (𝑏‘𝑗))
37	36, 27	oveq12d 7437	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑏‘𝑘) · (𝑤↑𝑘)) = ((𝑏‘𝑗) · (𝑤↑𝑗)))
38	37	cbvsumv 15771	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑤↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗))
39	35, 38	eqtrdi 2816	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
40	39	cbvmptv 5217	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑏‘𝑘) · (𝑧↑𝑘))) = (𝑤 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑛)((𝑏‘𝑗) · (𝑤↑𝑗)))
41	33, 40	eqtrdi 2816	. . . . . . . 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 592	. . . . . . . 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 26424	. . . . . . 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 1139	. . . . . 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 3224	. . . . 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 246	. . . 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 3224	. . 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 246	. 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 712	1 ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ (Poly‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∪ cun 3904   ⊆ wss 3906  {csn 4591   ↦ cmpt 5194   “ cima 5666  ‘cfv 6540  (class class class)co 7419   ∘_f cof 7682   ↑_m cmap 8830  ℂcc 11113  0cc0 11115  1c1 11116   + caddc 11118   · cmul 11120  ℕ₀cn0 12519  ℤ_≥cuz 12878  ...cfz 13551  ↑cexp 14115  Σcsu 15761  Polycply 26392

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-z 12607  df-uz 12879  df-rp 13033  df-fz 13552  df-fzo 13700  df-seq 14056  df-exp 14116  df-hash 14385  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-clim 15563  df-sum 15762  df-ply 26396

This theorem is used by:  plysub  26427  plymulcl  26429  plyco  26449  plyn0mulidp  26493  plydivlem2  26506  plydivlem4  26508  plydiveu  26510  mpaaeu  43935  rngunsnply  43954