MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coemullem Structured version   Visualization version   GIF version

Theorem coemullem 24999
Description: Lemma for coemul 25001 and dgrmul 25019. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1 𝐴 = (coeff‘𝐹)
coeadd.2 𝐵 = (coeff‘𝐺)
coeadd.3 𝑀 = (deg‘𝐹)
coeadd.4 𝑁 = (deg‘𝐺)
Assertion
Ref Expression
coemullem ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) ∧ (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁)))
Distinct variable groups:   𝑘,𝑛,𝐴   𝐵,𝑘,𝑛   𝑘,𝐹,𝑛   𝑘,𝑀   𝑘,𝐺,𝑛   𝑘,𝑁,𝑛   𝑆,𝑘,𝑛
Allowed substitution hint:   𝑀(𝑛)

Proof of Theorem coemullem
Dummy variables 𝑗 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plymulcl 24970 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
2 coeadd.3 . . . . 5 𝑀 = (deg‘𝐹)
3 dgrcl 24982 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
42, 3eqeltrid 2837 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
5 coeadd.4 . . . . 5 𝑁 = (deg‘𝐺)
6 dgrcl 24982 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
75, 6eqeltrid 2837 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
8 nn0addcl 12011 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
94, 7, 8syl2an 599 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ0)
10 fzfid 13432 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
11 coefv0.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
1211coef3 24981 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1312adantr 484 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
1413adantr 484 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ)
15 elfznn0 13091 . . . . . . 7 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
16 ffvelrn 6859 . . . . . . 7 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
1714, 15, 16syl2an 599 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴𝑘) ∈ ℂ)
18 coeadd.2 . . . . . . . . . 10 𝐵 = (coeff‘𝐺)
1918coef3 24981 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
2019adantl 485 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
2120ad2antrr 726 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ0⟶ℂ)
22 fznn0sub 13030 . . . . . . . 8 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
2322adantl 485 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛𝑘) ∈ ℕ0)
2421, 23ffvelrnd 6862 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛𝑘)) ∈ ℂ)
2517, 24mulcld 10739 . . . . 5 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2610, 25fsumcl 15183 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2726fmpttd 6889 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ)
28 oveq2 7178 . . . . . . . . . . 11 (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29 fvoveq1 7193 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐵‘(𝑛𝑘)) = (𝐵‘(𝑗𝑘)))
3029oveq2d 7186 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3130adantr 484 . . . . . . . . . . 11 ((𝑛 = 𝑗𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3228, 31sumeq12dv 15156 . . . . . . . . . 10 (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
33 eqid 2738 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))
34 sumex 15137 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) ∈ V
3532, 33, 34fvmpt 6775 . . . . . . . . 9 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3635ad2antrl 728 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
37 simp2r 1201 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ¬ 𝑗 ≤ (𝑀 + 𝑁))
38 simp2l 1200 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑗 ∈ ℕ0)
3938nn0red 12037 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑗 ∈ ℝ)
40 simp3l 1202 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ (0...𝑗))
41 elfznn0 13091 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ0)
4240, 41syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ ℕ0)
4342nn0red 12037 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ ℝ)
447adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
45443ad2ant1 1134 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℕ0)
4645nn0red 12037 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℝ)
4739, 43, 46lesubadd2d 11317 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑘 + 𝑁)))
484adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
49483ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑀 ∈ ℕ0)
5049nn0red 12037 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑀 ∈ ℝ)
51 simp3r 1203 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘𝑀)
5243, 50, 46, 51leadd1dd 11332 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁))
5343, 46readdcld 10748 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
5450, 46readdcld 10748 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55 letr 10812 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ (𝑘 + 𝑁) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁)))
5639, 53, 54, 55syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁)))
5752, 56mpan2d 694 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑗 ≤ (𝑘 + 𝑁) → 𝑗 ≤ (𝑀 + 𝑁)))
5847, 57sylbid 243 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑀 + 𝑁)))
5937, 58mtod 201 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ¬ (𝑗𝑘) ≤ 𝑁)
60 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
61603ad2ant1 1134 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝐺 ∈ (Poly‘𝑆))
62 fznn0sub 13030 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...𝑗) → (𝑗𝑘) ∈ ℕ0)
6340, 62syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑗𝑘) ∈ ℕ0)
6418, 5dgrub 24983 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗𝑘) ∈ ℕ0 ∧ (𝐵‘(𝑗𝑘)) ≠ 0) → (𝑗𝑘) ≤ 𝑁)
65643expia 1122 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗𝑘) ∈ ℕ0) → ((𝐵‘(𝑗𝑘)) ≠ 0 → (𝑗𝑘) ≤ 𝑁))
6661, 63, 65syl2anc 587 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐵‘(𝑗𝑘)) ≠ 0 → (𝑗𝑘) ≤ 𝑁))
6766necon1bd 2952 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (¬ (𝑗𝑘) ≤ 𝑁 → (𝐵‘(𝑗𝑘)) = 0))
6859, 67mpd 15 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝐵‘(𝑗𝑘)) = 0)
6968oveq2d 7186 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = ((𝐴𝑘) · 0))
70133ad2ant1 1134 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝐴:ℕ0⟶ℂ)
7170, 42ffvelrnd 6862 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝐴𝑘) ∈ ℂ)
7271mul01d 10917 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · 0) = 0)
7369, 72eqtrd 2773 . . . . . . . . . . . . 13 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
74733expia 1122 . . . . . . . . . . . 12 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0))
7574impl 459 . . . . . . . . . . 11 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
76 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
7776adantr 484 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆))
7811, 2dgrub 24983 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴𝑘) ≠ 0) → 𝑘𝑀)
79783expia 1122 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
8077, 41, 79syl2an 599 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
8180necon1bd 2952 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘𝑀 → (𝐴𝑘) = 0))
8281imp 410 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐴𝑘) = 0)
8382oveq1d 7185 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = (0 · (𝐵‘(𝑗𝑘))))
8420ad3antrrr 730 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → 𝐵:ℕ0⟶ℂ)
8562ad2antlr 727 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝑗𝑘) ∈ ℕ0)
8684, 85ffvelrnd 6862 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐵‘(𝑗𝑘)) ∈ ℂ)
8786mul02d 10916 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (0 · (𝐵‘(𝑗𝑘))) = 0)
8883, 87eqtrd 2773 . . . . . . . . . . 11 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
8975, 88pm2.61dan 813 . . . . . . . . . 10 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9089sumeq2dv 15153 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = Σ𝑘 ∈ (0...𝑗)0)
91 fzfi 13431 . . . . . . . . . . 11 (0...𝑗) ∈ Fin
9291olci 865 . . . . . . . . . 10 ((0...𝑗) ⊆ (ℤ‘0) ∨ (0...𝑗) ∈ Fin)
93 sumz 15172 . . . . . . . . . 10 (((0...𝑗) ⊆ (ℤ‘0) ∨ (0...𝑗) ∈ Fin) → Σ𝑘 ∈ (0...𝑗)0 = 0)
9492, 93ax-mp 5 . . . . . . . . 9 Σ𝑘 ∈ (0...𝑗)0 = 0
9590, 94eqtrdi 2789 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9636, 95eqtrd 2773 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0)
9796expr 460 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (¬ 𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0))
9897necon1ad 2951 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
9998ralrimiva 3096 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
100 plyco0 24941 . . . . 5 (((𝑀 + 𝑁) ∈ ℕ0 ∧ (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) “ (ℤ‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))))
1019, 27, 100syl2anc 587 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) “ (ℤ‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))))
10299, 101mpbird 260 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) “ (ℤ‘((𝑀 + 𝑁) + 1))) = {0})
10311, 2dgrub2 24984 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
104103adantr 484 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
10518, 5dgrub2 24984 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
106105adantl 485 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
10711, 2coeid 24987 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
108107adantr 484 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
10918, 5coeid 24987 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
110109adantl 485 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
11176, 60, 48, 44, 13, 20, 104, 106, 108, 110plymullem1 24963 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗))))
112 elfznn0 13091 . . . . . . . 8 (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ0)
113112, 35syl 17 . . . . . . 7 (𝑗 ∈ (0...(𝑀 + 𝑁)) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
114113oveq1d 7185 . . . . . 6 (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗)))
115114sumeq2i 15149 . . . . 5 Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗)) = Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗))
116115mpteq2i 5122 . . . 4 (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗)))
117111, 116eqtr4di 2791 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗))))
1181, 9, 27, 102, 117coeeq 24976 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))))
119 ffvelrn 6859 . . . 4 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ ∧ 𝑗 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ∈ ℂ)
12027, 112, 119syl2an 599 . . 3 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ∈ ℂ)
1211, 9, 120, 117dgrle 24992 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁))
122118, 121jca 515 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) ∧ (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wral 3053  wss 3843  {csn 4516   class class class wbr 5030  cmpt 5110  cima 5528  wf 6335  cfv 6339  (class class class)co 7170  f cof 7423  Fincfn 8555  cc 10613  cr 10614  0cc0 10615  1c1 10616   + caddc 10618   · cmul 10620  cle 10754  cmin 10948  0cn0 11976  cuz 12324  ...cfz 12981  cexp 13521  Σcsu 15135  Polycply 24933  coeffccoe 24935  degcdgr 24936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-n0 11977  df-z 12063  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-fl 13253  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-rlim 14936  df-sum 15136  df-0p 24422  df-ply 24937  df-coe 24939  df-dgr 24940
This theorem is referenced by:  coemul  25001  dgrmul2  25018
  Copyright terms: Public domain W3C validator