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Theorem coemullem 25411
Description: Lemma for coemul 25413 and dgrmul 25431. (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 25382 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
2 coeadd.3 . . . . 5 𝑀 = (deg‘𝐹)
3 dgrcl 25394 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
42, 3eqeltrid 2843 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
5 coeadd.4 . . . . 5 𝑁 = (deg‘𝐺)
6 dgrcl 25394 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
75, 6eqeltrid 2843 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
8 nn0addcl 12268 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
94, 7, 8syl2an 596 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ0)
10 fzfid 13693 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
11 coefv0.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
1211coef3 25393 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1312adantr 481 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
1413adantr 481 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ)
15 elfznn0 13349 . . . . . . 7 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
16 ffvelrn 6959 . . . . . . 7 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
1714, 15, 16syl2an 596 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴𝑘) ∈ ℂ)
18 coeadd.2 . . . . . . . . . 10 𝐵 = (coeff‘𝐺)
1918coef3 25393 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
2019adantl 482 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
2120ad2antrr 723 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ0⟶ℂ)
22 fznn0sub 13288 . . . . . . . 8 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
2322adantl 482 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛𝑘) ∈ ℕ0)
2421, 23ffvelrnd 6962 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛𝑘)) ∈ ℂ)
2517, 24mulcld 10995 . . . . 5 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2610, 25fsumcl 15445 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2726fmpttd 6989 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ)
28 oveq2 7283 . . . . . . . . . . 11 (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29 fvoveq1 7298 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐵‘(𝑛𝑘)) = (𝐵‘(𝑗𝑘)))
3029oveq2d 7291 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3130adantr 481 . . . . . . . . . . 11 ((𝑛 = 𝑗𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3228, 31sumeq12dv 15418 . . . . . . . . . 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 15399 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) ∈ V
3532, 33, 34fvmpt 6875 . . . . . . . . 9 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3635ad2antrl 725 . . . . . . . 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 1199 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ¬ 𝑗 ≤ (𝑀 + 𝑁))
38 simp2l 1198 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑗 ∈ ℕ0)
3938nn0red 12294 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑗 ∈ ℝ)
40 simp3l 1200 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ (0...𝑗))
41 elfznn0 13349 . . . . . . . . . . . . . . . . . . . . 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 12294 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ ℝ)
447adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
45443ad2ant1 1132 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℕ0)
4645nn0red 12294 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℝ)
4739, 43, 46lesubadd2d 11574 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑘 + 𝑁)))
484adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
49483ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑀 ∈ ℕ0)
5049nn0red 12294 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑀 ∈ ℝ)
51 simp3r 1201 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘𝑀)
5243, 50, 46, 51leadd1dd 11589 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁))
5343, 46readdcld 11004 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
5450, 46readdcld 11004 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55 letr 11069 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ (𝑘 + 𝑁) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁)))
5639, 53, 54, 55syl3anc 1370 . . . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑗 ≤ (𝑘 + 𝑁) → 𝑗 ≤ (𝑀 + 𝑁)))
5847, 57sylbid 239 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑀 + 𝑁)))
5937, 58mtod 197 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ¬ (𝑗𝑘) ≤ 𝑁)
60 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
61603ad2ant1 1132 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝐺 ∈ (Poly‘𝑆))
62 fznn0sub 13288 . . . . . . . . . . . . . . . . . . 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 25395 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗𝑘) ∈ ℕ0 ∧ (𝐵‘(𝑗𝑘)) ≠ 0) → (𝑗𝑘) ≤ 𝑁)
65643expia 1120 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗𝑘) ∈ ℕ0) → ((𝐵‘(𝑗𝑘)) ≠ 0 → (𝑗𝑘) ≤ 𝑁))
6661, 63, 65syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐵‘(𝑗𝑘)) ≠ 0 → (𝑗𝑘) ≤ 𝑁))
6766necon1bd 2961 . . . . . . . . . . . . . . . 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 7291 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = ((𝐴𝑘) · 0))
70133ad2ant1 1132 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝐴:ℕ0⟶ℂ)
7170, 42ffvelrnd 6962 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝐴𝑘) ∈ ℂ)
7271mul01d 11174 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · 0) = 0)
7369, 72eqtrd 2778 . . . . . . . . . . . . 13 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
74733expia 1120 . . . . . . . . . . . 12 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0))
7574impl 456 . . . . . . . . . . 11 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
76 simpl 483 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
7776adantr 481 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆))
7811, 2dgrub 25395 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴𝑘) ≠ 0) → 𝑘𝑀)
79783expia 1120 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
8077, 41, 79syl2an 596 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
8180necon1bd 2961 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘𝑀 → (𝐴𝑘) = 0))
8281imp 407 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐴𝑘) = 0)
8382oveq1d 7290 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = (0 · (𝐵‘(𝑗𝑘))))
8420ad3antrrr 727 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → 𝐵:ℕ0⟶ℂ)
8562ad2antlr 724 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝑗𝑘) ∈ ℕ0)
8684, 85ffvelrnd 6962 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐵‘(𝑗𝑘)) ∈ ℂ)
8786mul02d 11173 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (0 · (𝐵‘(𝑗𝑘))) = 0)
8883, 87eqtrd 2778 . . . . . . . . . . 11 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
8975, 88pm2.61dan 810 . . . . . . . . . 10 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9089sumeq2dv 15415 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = Σ𝑘 ∈ (0...𝑗)0)
91 fzfi 13692 . . . . . . . . . . 11 (0...𝑗) ∈ Fin
9291olci 863 . . . . . . . . . 10 ((0...𝑗) ⊆ (ℤ‘0) ∨ (0...𝑗) ∈ Fin)
93 sumz 15434 . . . . . . . . . 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 2794 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9636, 95eqtrd 2778 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0)
9796expr 457 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (¬ 𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0))
9897necon1ad 2960 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
9998ralrimiva 3103 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
100 plyco0 25353 . . . . 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 584 . . . 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 256 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) “ (ℤ‘((𝑀 + 𝑁) + 1))) = {0})
10311, 2dgrub2 25396 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
104103adantr 481 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
10518, 5dgrub2 25396 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
106105adantl 482 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
10711, 2coeid 25399 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
108107adantr 481 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
10918, 5coeid 25399 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
110109adantl 482 . . . . 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 25375 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗))))
112 elfznn0 13349 . . . . . . . 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 7290 . . . . . 6 (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗)))
115114sumeq2i 15411 . . . . 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 5179 . . . 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 2796 . . 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 25388 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))))
119 ffvelrn 6959 . . . 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 596 . . 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 25404 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁))
122118, 121jca 512 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 205  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wral 3064  wss 3887  {csn 4561   class class class wbr 5074  cmpt 5157  cima 5592  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  Fincfn 8733  cc 10869  cr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  cle 11010  cmin 11205  0cn0 12233  cuz 12582  ...cfz 13239  cexp 13782  Σcsu 15397  Polycply 25345  coeffccoe 25347  degcdgr 25348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-0p 24834  df-ply 25349  df-coe 25351  df-dgr 25352
This theorem is referenced by:  coemul  25413  dgrmul2  25430
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