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

Theorem coemullem 26303
Description: Lemma for coemul 26305 and dgrmul 26324. (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 26274 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
2 coeadd.3 . . . . 5 𝑀 = (deg‘𝐹)
3 dgrcl 26286 . . . . 5 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
42, 3eqeltrid 2842 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
5 coeadd.4 . . . . 5 𝑁 = (deg‘𝐺)
6 dgrcl 26286 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
75, 6eqeltrid 2842 . . . 4 (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
8 nn0addcl 12558 . . . 4 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
94, 7, 8syl2an 596 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ0)
10 fzfid 14010 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
11 coefv0.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
1211coef3 26285 . . . . . . . . 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 13656 . . . . . . 7 (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
16 ffvelcdm 7100 . . . . . . 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 26285 . . . . . . . . 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 13592 . . . . . . . 8 (𝑘 ∈ (0...𝑛) → (𝑛𝑘) ∈ ℕ0)
2322adantl 481 . . . . . . 7 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛𝑘) ∈ ℕ0)
2421, 23ffvelcdmd 7104 . . . . . 6 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛𝑘)) ∈ ℂ)
2517, 24mulcld 11278 . . . . 5 ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2610, 25fsumcl 15765 . . . 4 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) ∈ ℂ)
2726fmpttd 7134 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))):ℕ0⟶ℂ)
28 oveq2 7438 . . . . . . . . . . 11 (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29 fvoveq1 7453 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (𝐵‘(𝑛𝑘)) = (𝐵‘(𝑗𝑘)))
3029oveq2d 7446 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3130adantr 480 . . . . . . . . . . 11 ((𝑛 = 𝑗𝑘 ∈ (0...𝑛)) → ((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = ((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
3228, 31sumeq12dv 15738 . . . . . . . . . 10 (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))))
33 eqid 2734 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))
34 sumex 15720 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) ∈ V
3532, 33, 34fvmpt 7015 . . . . . . . . 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 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 12585 . . . . . . . . . . . . . . . . . . 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 13656 . . . . . . . . . . . . . . . . . . . . 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 12585 . . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℕ0)
4645nn0red 12585 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑁 ∈ ℝ)
4739, 43, 46lesubadd2d 11859 . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝑀 ∈ ℕ0)
5049nn0red 12585 . . . . . . . . . . . . . . . . . . . 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 11874 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁))
5343, 46readdcld 11287 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
5450, 46readdcld 11287 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55 letr 11352 . . . . . . . . . . . . . . . . . . . 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 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 1132 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → 𝐺 ∈ (Poly‘𝑆))
62 fznn0sub 13592 . . . . . . . . . . . . . . . . . . 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 26287 . . . . . . . . . . . . . . . . . . 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 2955 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 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, 42ffvelcdmd 7104 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → (𝐴𝑘) ∈ ℂ)
7271mul01d 11457 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘𝑀)) → ((𝐴𝑘) · 0) = 0)
7369, 72eqtrd 2774 . . . . . . . . . . . . 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 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 26287 . . . . . . . . . . . . . . . . 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 2955 . . . . . . . . . . . . . 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 7445 . . . . . . . . . . . 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 7104 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (𝐵‘(𝑗𝑘)) ∈ ℂ)
8786mul02d 11456 . . . . . . . . . . . 12 (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘𝑀) → (0 · (𝐵‘(𝑗𝑘))) = 0)
8883, 87eqtrd 2774 . . . . . . . . . . 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 15734 . . . . . . . . 9 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = Σ𝑘 ∈ (0...𝑗)0)
91 fzfi 14009 . . . . . . . . . . 11 (0...𝑗) ∈ Fin
9291olci 866 . . . . . . . . . 10 ((0...𝑗) ⊆ (ℤ‘0) ∨ (0...𝑗) ∈ Fin)
93 sumz 15754 . . . . . . . . . 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 2790 . . . . . . . 8 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) = 0)
9636, 95eqtrd 2774 . . . . . . 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 2954 . . . . 5 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
9998ralrimiva 3143 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
100 plyco0 26245 . . . . 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 26288 . . . . . 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 26288 . . . . . 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 26291 . . . . . 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 26291 . . . . . 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 26267 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗))))
112 elfznn0 13656 . . . . . . . 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 7445 . . . . . 6 (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))))‘𝑗) · (𝑧𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴𝑘) · (𝐵‘(𝑗𝑘))) · (𝑧𝑗)))
115114sumeq2i 15730 . . . . 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 5252 . . . 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 2792 . . 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 26280 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))))
119 ffvelcdm 7100 . . . 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 26296 . 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 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wss 3962  {csn 4630   class class class wbr 5147  cmpt 5230  cima 5691  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  cle 11293  cmin 11489  0cn0 12523  cuz 12875  ...cfz 13543  cexp 14098  Σcsu 15718  Polycply 26237  coeffccoe 26239  degcdgr 26240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-0p 25718  df-ply 26241  df-coe 26243  df-dgr 26244
This theorem is referenced by:  coemul  26305  dgrmul2  26323
  Copyright terms: Public domain W3C validator