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Theorem coemullem 26289
Description: Lemma for coemul 26291 and dgrmul 26310. (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 26260 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
2 coeadd.3 . . . . 5 𝑀 = (deg‘𝐹)
3 dgrcl 26272 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
42, 3eqeltrid 2845 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
5 coeadd.4 . . . . 5 𝑁 = (deg‘𝐺)
6 dgrcl 26272 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
75, 6eqeltrid 2845 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
8 nn0addcl 12561 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
94, 7, 8syl2an 596 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ0)
10 fzfid 14014 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
11 coefv0.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
1211coef3 26271 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
1312adantr 480 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
1413adantr 480 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ)
15 elfznn0 13660 . . . . . . 7 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
16 ffvelcdm 7101 . . . . . . 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 26271 . . . . . . . . 9 (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
2019adantl 481 . . . . . . . 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 13596 . . . . . . . 8 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
2322adantl 481 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛𝑘) ∈ ℕ0)
2421, 23ffvelcdmd 7105 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛𝑘)) ∈ ℂ)
2517, 24mulcld 11281 . . . . 5 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2610, 25fsumcl 15769 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2726fmpttd 7135 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ)
28 oveq2 7439 . . . . . . . . . . 11 (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29 fvoveq1 7454 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐵‘(𝑛𝑘)) = (𝐵‘(𝑗𝑘)))
3029oveq2d 7447 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3130adantr 480 . . . . . . . . . . 11 ((𝑛 = 𝑗𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3228, 31sumeq12dv 15742 . . . . . . . . . 10 (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
33 eqid 2737 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))
34 sumex 15724 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) ∈ V
3532, 33, 34fvmpt 7016 . . . . . . . . 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 12588 . . . . . . . . . . . . . . . . . . 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 13660 . . . . . . . . . . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑘 ∈ ℝ)
447adantl 481 . . . . . . . . . . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℝ)
4739, 43, 46lesubadd2d 11862 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑘 + 𝑁)))
484adantr 480 . . . . . . . . . . . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . . . . . . 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 11877 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁))
5343, 46readdcld 11290 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
5450, 46readdcld 11290 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55 letr 11355 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ (𝑘 + 𝑁) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁)))
5639, 53, 54, 55syl3anc 1373 . . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝑗𝑘) ≤ 𝑁𝑗 ≤ (𝑀 + 𝑁)))
5937, 58mtod 198 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ¬ (𝑗𝑘) ≤ 𝑁)
60 simpr 484 . . . . . . . . . . . . . . . . . . 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 13596 . . . . . . . . . . . . . . . . . . 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 26273 . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐵‘(𝑗𝑘)) ≠ 0 → (𝑗𝑘) ≤ 𝑁))
6766necon1bd 2958 . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝐴𝑘) ∈ ℂ)
7271mul01d 11460 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · 0) = 0)
7369, 72eqtrd 2777 . . . . . . . . . . . . 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 455 . . . . . . . . . . 11 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘𝑀) → ((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
76 simpl 482 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
7776adantr 480 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆))
7811, 2dgrub 26273 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴𝑘) ≠ 0) → 𝑘𝑀)
79783expia 1122 . . . . . . . . . . . . . . . 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 2958 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘𝑀 → (𝐴𝑘) = 0))
8281imp 406 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐴𝑘) = 0)
8382oveq1d 7446 . . . . . . . . . . . 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, 85ffvelcdmd 7105 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐵‘(𝑗𝑘)) ∈ ℂ)
8786mul02d 11459 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (0 · (𝐵‘(𝑗𝑘))) = 0)
8883, 87eqtrd 2777 . . . . . . . . . . 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 15738 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = Σ𝑘 ∈ (0...𝑗)0)
91 fzfi 14013 . . . . . . . . . . 11 (0...𝑗) ∈ Fin
9291olci 867 . . . . . . . . . 10 ((0...𝑗) ⊆ (ℤ‘0) ∨ (0...𝑗) ∈ Fin)
93 sumz 15758 . . . . . . . . . 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 2793 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9636, 95eqtrd 2777 . . . . . . 7 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0)
9796expr 456 . . . . . 6 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (¬ 𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) = 0))
9897necon1ad 2957 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
9998ralrimiva 3146 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
100 plyco0 26231 . . . . 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 257 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) “ (ℤ‘((𝑀 + 𝑁) + 1))) = {0})
10311, 2dgrub2 26274 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
104103adantr 480 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})
10518, 5dgrub2 26274 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
106105adantl 481 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})
10711, 2coeid 26277 . . . . . 6 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
108107adantr 480 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))
10918, 5coeid 26277 . . . . . 6 (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))
110109adantl 481 . . . . 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 26253 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗))))
112 elfznn0 13660 . . . . . . . 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 7446 . . . . . 6 (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗)))
115114sumeq2i 15734 . . . . 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 5247 . . . 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 2795 . . 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 26266 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))))
119 ffvelcdm 7101 . . . 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 26282 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁))
122118, 121jca 511 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wss 3951  {csn 4626   class class class wbr 5143  cmpt 5225  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  cle 11296  cmin 11492  0cn0 12526  cuz 12878  ...cfz 13547  cexp 14102  Σcsu 15722  Polycply 26223  coeffccoe 26225  degcdgr 26226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-coe 26229  df-dgr 26230
This theorem is referenced by:  coemul  26291  dgrmul2  26309
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