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Theorem coemullem 25733
Description: Lemma for coemul 25735 and dgrmul 25753. (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 25704 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
2 coeadd.3 . . . . 5 𝑀 = (deg‘𝐹)
3 dgrcl 25716 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
42, 3eqeltrid 2838 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
5 coeadd.4 . . . . 5 𝑁 = (deg‘𝐺)
6 dgrcl 25716 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
75, 6eqeltrid 2838 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
8 nn0addcl 12494 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
94, 7, 8syl2an 597 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ0)
10 fzfid 13925 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
11 coefv0.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
1211coef3 25715 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1312adantr 482 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
1413adantr 482 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ)
15 elfznn0 13581 . . . . . . 7 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
16 ffvelcdm 7071 . . . . . . 7 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
1714, 15, 16syl2an 597 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴𝑘) ∈ ℂ)
18 coeadd.2 . . . . . . . . . 10 𝐵 = (coeff‘𝐺)
1918coef3 25715 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
2019adantl 483 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
2120ad2antrr 725 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ0⟶ℂ)
22 fznn0sub 13520 . . . . . . . 8 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
2322adantl 483 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛𝑘) ∈ ℕ0)
2421, 23ffvelcdmd 7075 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛𝑘)) ∈ ℂ)
2517, 24mulcld 11221 . . . . 5 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2610, 25fsumcl 15666 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2726fmpttd 7102 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ)
28 oveq2 7404 . . . . . . . . . . 11 (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29 fvoveq1 7419 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐵‘(𝑛𝑘)) = (𝐵‘(𝑗𝑘)))
3029oveq2d 7412 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3130adantr 482 . . . . . . . . . . 11 ((𝑛 = 𝑗𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3228, 31sumeq12dv 15639 . . . . . . . . . 10 (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
33 eqid 2733 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))
34 sumex 15621 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) ∈ V
3532, 33, 34fvmpt 6987 . . . . . . . . 9 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3635ad2antrl 727 . . . . . . . 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 12520 . . . . . . . . . . . . . . . . . . 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 13581 . . . . . . . . . . . . . . . . . . . . 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 12520 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ ℝ)
447adantl 483 . . . . . . . . . . . . . . . . . . . . 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 12520 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℝ)
4739, 43, 46lesubadd2d 11800 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑘 + 𝑁)))
484adantr 482 . . . . . . . . . . . . . . . . . . . . . 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 12520 . . . . . . . . . . . . . . . . . . . 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 11815 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁))
5343, 46readdcld 11230 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
5450, 46readdcld 11230 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55 letr 11295 . . . . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . 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 13520 . . . . . . . . . . . . . . . . . . 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 25717 . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐵‘(𝑗𝑘)) ≠ 0 → (𝑗𝑘) ≤ 𝑁))
6766necon1bd 2959 . . . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . 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, 42ffvelcdmd 7075 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝐴𝑘) ∈ ℂ)
7271mul01d 11400 . . . . . . . . . . . . . 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 457 . . . . . . . . . . 11 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
76 simpl 484 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
7776adantr 482 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆))
7811, 2dgrub 25717 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴𝑘) ≠ 0) → 𝑘𝑀)
79783expia 1122 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
8077, 41, 79syl2an 597 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴𝑘) ≠ 0 → 𝑘𝑀))
8180necon1bd 2959 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘𝑀 → (𝐴𝑘) = 0))
8281imp 408 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐴𝑘) = 0)
8382oveq1d 7411 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = (0 · (𝐵‘(𝑗𝑘))))
8420ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → 𝐵:ℕ0⟶ℂ)
8562ad2antlr 726 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝑗𝑘) ∈ ℕ0)
8684, 85ffvelcdmd 7075 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐵‘(𝑗𝑘)) ∈ ℂ)
8786mul02d 11399 . . . . . . . . . . . 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 812 . . . . . . . . . 10 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9089sumeq2dv 15636 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = Σ𝑘 ∈ (0...𝑗)0)
91 fzfi 13924 . . . . . . . . . . 11 (0...𝑗) ∈ Fin
9291olci 865 . . . . . . . . . 10 ((0...𝑗) ⊆ (ℤ‘0) ∨ (0...𝑗) ∈ Fin)
93 sumz 15655 . . . . . . . . . 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 458 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (¬ 𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0))
9897necon1ad 2958 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
9998ralrimiva 3147 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
100 plyco0 25675 . . . . 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 585 . . . 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 257 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) “ (ℤ‘((𝑀 + 𝑁) + 1))) = {0})
10311, 2dgrub2 25718 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
104103adantr 482 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
10518, 5dgrub2 25718 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
106105adantl 483 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
10711, 2coeid 25721 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
108107adantr 482 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
10918, 5coeid 25721 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
110109adantl 483 . . . . 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 25697 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗))))
112 elfznn0 13581 . . . . . . . 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 7411 . . . . . 6 (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗)))
115114sumeq2i 15632 . . . . 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 5249 . . . 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 25710 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))))
119 ffvelcdm 7071 . . . 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 597 . . 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 25726 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁))
122118, 121jca 513 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  wss 3946  {csn 4624   class class class wbr 5144  cmpt 5227  cima 5675  wf 6531  cfv 6535  (class class class)co 7396  f cof 7655  Fincfn 8927  cc 11095  cr 11096  0cc0 11097  1c1 11098   + caddc 11100   · cmul 11102  cle 11236  cmin 11431  0cn0 12459  cuz 12809  ...cfz 13471  cexp 14014  Σcsu 15619  Polycply 25667  coeffccoe 25669  degcdgr 25670
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-inf2 9623  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174  ax-pre-sup 11175
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  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 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9424  df-inf 9425  df-oi 9492  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-div 11859  df-nn 12200  df-2 12262  df-3 12263  df-n0 12460  df-z 12546  df-uz 12810  df-rp 12962  df-fz 13472  df-fzo 13615  df-fl 13744  df-seq 13954  df-exp 14015  df-hash 14278  df-cj 15033  df-re 15034  df-im 15035  df-sqrt 15169  df-abs 15170  df-clim 15419  df-rlim 15420  df-sum 15620  df-0p 25156  df-ply 25671  df-coe 25673  df-dgr 25674
This theorem is referenced by:  coemul  25735  dgrmul2  25752
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