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Theorem coemullem 26103
Description: Lemma for coemul 26105 and dgrmul 26124. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1 𝐴 = (coeffβ€˜πΉ)
coeadd.2 𝐡 = (coeffβ€˜πΊ)
coeadd.3 𝑀 = (degβ€˜πΉ)
coeadd.4 𝑁 = (degβ€˜πΊ)
Assertion
Ref Expression
coemullem ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ ((coeffβ€˜(𝐹 ∘f Β· 𝐺)) = (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))) ∧ (degβ€˜(𝐹 ∘f Β· 𝐺)) ≀ (𝑀 + 𝑁)))
Distinct variable groups:   π‘˜,𝑛,𝐴   𝐡,π‘˜,𝑛   π‘˜,𝐹,𝑛   π‘˜,𝑀   π‘˜,𝐺,𝑛   π‘˜,𝑁,𝑛   𝑆,π‘˜,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 26074 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f Β· 𝐺) ∈ (Polyβ€˜β„‚))
2 coeadd.3 . . . . 5 𝑀 = (degβ€˜πΉ)
3 dgrcl 26086 . . . . 5 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΉ) ∈ β„•0)
42, 3eqeltrid 2829 . . . 4 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝑀 ∈ β„•0)
5 coeadd.4 . . . . 5 𝑁 = (degβ€˜πΊ)
6 dgrcl 26086 . . . . 5 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΊ) ∈ β„•0)
75, 6eqeltrid 2829 . . . 4 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝑁 ∈ β„•0)
8 nn0addcl 12503 . . . 4 ((𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝑀 + 𝑁) ∈ β„•0)
94, 7, 8syl2an 595 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝑀 + 𝑁) ∈ β„•0)
10 fzfid 13934 . . . . 5 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) β†’ (0...𝑛) ∈ Fin)
11 coefv0.1 . . . . . . . . . 10 𝐴 = (coeffβ€˜πΉ)
1211coef3 26085 . . . . . . . . 9 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐴:β„•0βŸΆβ„‚)
1312adantr 480 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐴:β„•0βŸΆβ„‚)
1413adantr 480 . . . . . . 7 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) β†’ 𝐴:β„•0βŸΆβ„‚)
15 elfznn0 13590 . . . . . . 7 (π‘˜ ∈ (0...𝑛) β†’ π‘˜ ∈ β„•0)
16 ffvelcdm 7073 . . . . . . 7 ((𝐴:β„•0βŸΆβ„‚ ∧ π‘˜ ∈ β„•0) β†’ (π΄β€˜π‘˜) ∈ β„‚)
1714, 15, 16syl2an 595 . . . . . 6 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) ∧ π‘˜ ∈ (0...𝑛)) β†’ (π΄β€˜π‘˜) ∈ β„‚)
18 coeadd.2 . . . . . . . . . 10 𝐡 = (coeffβ€˜πΊ)
1918coef3 26085 . . . . . . . . 9 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝐡:β„•0βŸΆβ„‚)
2019adantl 481 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐡:β„•0βŸΆβ„‚)
2120ad2antrr 723 . . . . . . 7 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) ∧ π‘˜ ∈ (0...𝑛)) β†’ 𝐡:β„•0βŸΆβ„‚)
22 fznn0sub 13529 . . . . . . . 8 (π‘˜ ∈ (0...𝑛) β†’ (𝑛 βˆ’ π‘˜) ∈ β„•0)
2322adantl 481 . . . . . . 7 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) ∧ π‘˜ ∈ (0...𝑛)) β†’ (𝑛 βˆ’ π‘˜) ∈ β„•0)
2421, 23ffvelcdmd 7077 . . . . . 6 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) ∧ π‘˜ ∈ (0...𝑛)) β†’ (π΅β€˜(𝑛 βˆ’ π‘˜)) ∈ β„‚)
2517, 24mulcld 11230 . . . . 5 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) ∧ π‘˜ ∈ (0...𝑛)) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))) ∈ β„‚)
2610, 25fsumcl 15675 . . . 4 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑛 ∈ β„•0) β†’ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))) ∈ β„‚)
2726fmpttd 7106 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))):β„•0βŸΆβ„‚)
28 oveq2 7409 . . . . . . . . . . 11 (𝑛 = 𝑗 β†’ (0...𝑛) = (0...𝑗))
29 fvoveq1 7424 . . . . . . . . . . . . 13 (𝑛 = 𝑗 β†’ (π΅β€˜(𝑛 βˆ’ π‘˜)) = (π΅β€˜(𝑗 βˆ’ π‘˜)))
3029oveq2d 7417 . . . . . . . . . . . 12 (𝑛 = 𝑗 β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))) = ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
3130adantr 480 . . . . . . . . . . 11 ((𝑛 = 𝑗 ∧ π‘˜ ∈ (0...𝑛)) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))) = ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
3228, 31sumeq12dv 15648 . . . . . . . . . 10 (𝑛 = 𝑗 β†’ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))) = Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
33 eqid 2724 . . . . . . . . . 10 (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))) = (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))
34 sumex 15630 . . . . . . . . . 10 Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) ∈ V
3532, 33, 34fvmpt 6988 . . . . . . . . 9 (𝑗 ∈ β„•0 β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) = Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
3635ad2antrl 725 . . . . . . . 8 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) = Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
37 simp2r 1197 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ Β¬ 𝑗 ≀ (𝑀 + 𝑁))
38 simp2l 1196 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝑗 ∈ β„•0)
3938nn0red 12529 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝑗 ∈ ℝ)
40 simp3l 1198 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ π‘˜ ∈ (0...𝑗))
41 elfznn0 13590 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ ∈ (0...𝑗) β†’ π‘˜ ∈ β„•0)
4240, 41syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ π‘˜ ∈ β„•0)
4342nn0red 12529 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ π‘˜ ∈ ℝ)
447adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝑁 ∈ β„•0)
45443ad2ant1 1130 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝑁 ∈ β„•0)
4645nn0red 12529 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝑁 ∈ ℝ)
4739, 43, 46lesubadd2d 11809 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((𝑗 βˆ’ π‘˜) ≀ 𝑁 ↔ 𝑗 ≀ (π‘˜ + 𝑁)))
484adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝑀 ∈ β„•0)
49483ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝑀 ∈ β„•0)
5049nn0red 12529 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝑀 ∈ ℝ)
51 simp3r 1199 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ π‘˜ ≀ 𝑀)
5243, 50, 46, 51leadd1dd 11824 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (π‘˜ + 𝑁) ≀ (𝑀 + 𝑁))
5343, 46readdcld 11239 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (π‘˜ + 𝑁) ∈ ℝ)
5450, 46readdcld 11239 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (𝑀 + 𝑁) ∈ ℝ)
55 letr 11304 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ (π‘˜ + 𝑁) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) β†’ ((𝑗 ≀ (π‘˜ + 𝑁) ∧ (π‘˜ + 𝑁) ≀ (𝑀 + 𝑁)) β†’ 𝑗 ≀ (𝑀 + 𝑁)))
5639, 53, 54, 55syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((𝑗 ≀ (π‘˜ + 𝑁) ∧ (π‘˜ + 𝑁) ≀ (𝑀 + 𝑁)) β†’ 𝑗 ≀ (𝑀 + 𝑁)))
5752, 56mpan2d 691 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (𝑗 ≀ (π‘˜ + 𝑁) β†’ 𝑗 ≀ (𝑀 + 𝑁)))
5847, 57sylbid 239 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((𝑗 βˆ’ π‘˜) ≀ 𝑁 β†’ 𝑗 ≀ (𝑀 + 𝑁)))
5937, 58mtod 197 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ Β¬ (𝑗 βˆ’ π‘˜) ≀ 𝑁)
60 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
61603ad2ant1 1130 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
62 fznn0sub 13529 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ (0...𝑗) β†’ (𝑗 βˆ’ π‘˜) ∈ β„•0)
6340, 62syl 17 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (𝑗 βˆ’ π‘˜) ∈ β„•0)
6418, 5dgrub 26087 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ (Polyβ€˜π‘†) ∧ (𝑗 βˆ’ π‘˜) ∈ β„•0 ∧ (π΅β€˜(𝑗 βˆ’ π‘˜)) β‰  0) β†’ (𝑗 βˆ’ π‘˜) ≀ 𝑁)
65643expia 1118 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ (Polyβ€˜π‘†) ∧ (𝑗 βˆ’ π‘˜) ∈ β„•0) β†’ ((π΅β€˜(𝑗 βˆ’ π‘˜)) β‰  0 β†’ (𝑗 βˆ’ π‘˜) ≀ 𝑁))
6661, 63, 65syl2anc 583 . . . . . . . . . . . . . . . . 17 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((π΅β€˜(𝑗 βˆ’ π‘˜)) β‰  0 β†’ (𝑗 βˆ’ π‘˜) ≀ 𝑁))
6766necon1bd 2950 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (Β¬ (𝑗 βˆ’ π‘˜) ≀ 𝑁 β†’ (π΅β€˜(𝑗 βˆ’ π‘˜)) = 0))
6859, 67mpd 15 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (π΅β€˜(𝑗 βˆ’ π‘˜)) = 0)
6968oveq2d 7417 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = ((π΄β€˜π‘˜) Β· 0))
70133ad2ant1 1130 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ 𝐴:β„•0βŸΆβ„‚)
7170, 42ffvelcdmd 7077 . . . . . . . . . . . . . . 15 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ (π΄β€˜π‘˜) ∈ β„‚)
7271mul01d 11409 . . . . . . . . . . . . . 14 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((π΄β€˜π‘˜) Β· 0) = 0)
7369, 72eqtrd 2764 . . . . . . . . . . . . 13 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁)) ∧ (π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀)) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0)
74733expia 1118 . . . . . . . . . . . 12 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) β†’ ((π‘˜ ∈ (0...𝑗) ∧ π‘˜ ≀ 𝑀) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0))
7574impl 455 . . . . . . . . . . 11 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ π‘˜ ≀ 𝑀) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0)
76 simpl 482 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
7776adantr 480 . . . . . . . . . . . . . . . 16 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
7811, 2dgrub 26087 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ π‘˜ ∈ β„•0 ∧ (π΄β€˜π‘˜) β‰  0) β†’ π‘˜ ≀ 𝑀)
79783expia 1118 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ π‘˜ ∈ β„•0) β†’ ((π΄β€˜π‘˜) β‰  0 β†’ π‘˜ ≀ 𝑀))
8077, 41, 79syl2an 595 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) β†’ ((π΄β€˜π‘˜) β‰  0 β†’ π‘˜ ≀ 𝑀))
8180necon1bd 2950 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) β†’ (Β¬ π‘˜ ≀ 𝑀 β†’ (π΄β€˜π‘˜) = 0))
8281imp 406 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ (π΄β€˜π‘˜) = 0)
8382oveq1d 7416 . . . . . . . . . . . 12 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = (0 Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
8420ad3antrrr 727 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ 𝐡:β„•0βŸΆβ„‚)
8562ad2antlr 724 . . . . . . . . . . . . . 14 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ (𝑗 βˆ’ π‘˜) ∈ β„•0)
8684, 85ffvelcdmd 7077 . . . . . . . . . . . . 13 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ (π΅β€˜(𝑗 βˆ’ π‘˜)) ∈ β„‚)
8786mul02d 11408 . . . . . . . . . . . 12 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ (0 Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0)
8883, 87eqtrd 2764 . . . . . . . . . . 11 (((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) ∧ Β¬ π‘˜ ≀ 𝑀) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0)
8975, 88pm2.61dan 810 . . . . . . . . . 10 ((((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) ∧ π‘˜ ∈ (0...𝑗)) β†’ ((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0)
9089sumeq2dv 15645 . . . . . . . . 9 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) β†’ Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = Ξ£π‘˜ ∈ (0...𝑗)0)
91 fzfi 13933 . . . . . . . . . . 11 (0...𝑗) ∈ Fin
9291olci 863 . . . . . . . . . 10 ((0...𝑗) βŠ† (β„€β‰₯β€˜0) ∨ (0...𝑗) ∈ Fin)
93 sumz 15664 . . . . . . . . . 10 (((0...𝑗) βŠ† (β„€β‰₯β€˜0) ∨ (0...𝑗) ∈ Fin) β†’ Ξ£π‘˜ ∈ (0...𝑗)0 = 0)
9492, 93ax-mp 5 . . . . . . . . 9 Ξ£π‘˜ ∈ (0...𝑗)0 = 0
9590, 94eqtrdi 2780 . . . . . . . 8 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) β†’ Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) = 0)
9636, 95eqtrd 2764 . . . . . . 7 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ (𝑗 ∈ β„•0 ∧ Β¬ 𝑗 ≀ (𝑀 + 𝑁))) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) = 0)
9796expr 456 . . . . . 6 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑗 ∈ β„•0) β†’ (Β¬ 𝑗 ≀ (𝑀 + 𝑁) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) = 0))
9897necon1ad 2949 . . . . 5 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑗 ∈ β„•0) β†’ (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) β‰  0 β†’ 𝑗 ≀ (𝑀 + 𝑁)))
9998ralrimiva 3138 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ βˆ€π‘— ∈ β„•0 (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) β‰  0 β†’ 𝑗 ≀ (𝑀 + 𝑁)))
100 plyco0 26045 . . . . 5 (((𝑀 + 𝑁) ∈ β„•0 ∧ (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))):β„•0βŸΆβ„‚) β†’ (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))) β€œ (β„€β‰₯β€˜((𝑀 + 𝑁) + 1))) = {0} ↔ βˆ€π‘— ∈ β„•0 (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) β‰  0 β†’ 𝑗 ≀ (𝑀 + 𝑁))))
1019, 27, 100syl2anc 583 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))) β€œ (β„€β‰₯β€˜((𝑀 + 𝑁) + 1))) = {0} ↔ βˆ€π‘— ∈ β„•0 (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) β‰  0 β†’ 𝑗 ≀ (𝑀 + 𝑁))))
10299, 101mpbird 257 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))) β€œ (β„€β‰₯β€˜((𝑀 + 𝑁) + 1))) = {0})
10311, 2dgrub2 26088 . . . . . 6 (𝐹 ∈ (Polyβ€˜π‘†) β†’ (𝐴 β€œ (β„€β‰₯β€˜(𝑀 + 1))) = {0})
104103adantr 480 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐴 β€œ (β„€β‰₯β€˜(𝑀 + 1))) = {0})
10518, 5dgrub2 26088 . . . . . 6 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (𝐡 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0})
106105adantl 481 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐡 β€œ (β„€β‰₯β€˜(𝑁 + 1))) = {0})
10711, 2coeid 26091 . . . . . 6 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑀)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
108107adantr 480 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐹 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑀)((π΄β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
10918, 5coeid 26091 . . . . . 6 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝐺 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΅β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
110109adantl 481 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐺 = (𝑧 ∈ β„‚ ↦ Ξ£π‘˜ ∈ (0...𝑁)((π΅β€˜π‘˜) Β· (π‘§β†‘π‘˜))))
11176, 60, 48, 44, 13, 20, 104, 106, 108, 110plymullem1 26067 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f Β· 𝐺) = (𝑧 ∈ β„‚ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) Β· (𝑧↑𝑗))))
112 elfznn0 13590 . . . . . . . 8 (𝑗 ∈ (0...(𝑀 + 𝑁)) β†’ 𝑗 ∈ β„•0)
113112, 35syl 17 . . . . . . 7 (𝑗 ∈ (0...(𝑀 + 𝑁)) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) = Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))))
114113oveq1d 7416 . . . . . 6 (𝑗 ∈ (0...(𝑀 + 𝑁)) β†’ (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) Β· (𝑧↑𝑗)) = (Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) Β· (𝑧↑𝑗)))
115114sumeq2i 15641 . . . . 5 Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) Β· (𝑧↑𝑗)) = Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) Β· (𝑧↑𝑗))
116115mpteq2i 5243 . . . 4 (𝑧 ∈ β„‚ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) Β· (𝑧↑𝑗))) = (𝑧 ∈ β„‚ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Ξ£π‘˜ ∈ (0...𝑗)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑗 βˆ’ π‘˜))) Β· (𝑧↑𝑗)))
117111, 116eqtr4di 2782 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f Β· 𝐺) = (𝑧 ∈ β„‚ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) Β· (𝑧↑𝑗))))
1181, 9, 27, 102, 117coeeq 26080 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (coeffβ€˜(𝐹 ∘f Β· 𝐺)) = (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))))
119 ffvelcdm 7073 . . . 4 (((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))):β„•0βŸΆβ„‚ ∧ 𝑗 ∈ β„•0) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) ∈ β„‚)
12027, 112, 119syl2an 595 . . 3 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) β†’ ((𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜))))β€˜π‘—) ∈ β„‚)
1211, 9, 120, 117dgrle 26096 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (degβ€˜(𝐹 ∘f Β· 𝐺)) ≀ (𝑀 + 𝑁))
122118, 121jca 511 1 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ ((coeffβ€˜(𝐹 ∘f Β· 𝐺)) = (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β€˜π‘˜) Β· (π΅β€˜(𝑛 βˆ’ π‘˜)))) ∧ (degβ€˜(𝐹 ∘f Β· 𝐺)) ≀ (𝑀 + 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053   βŠ† wss 3940  {csn 4620   class class class wbr 5138   ↦ cmpt 5221   β€œ cima 5669  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ∘f cof 7661  Fincfn 8934  β„‚cc 11103  β„cr 11104  0cc0 11105  1c1 11106   + caddc 11108   Β· cmul 11110   ≀ cle 11245   βˆ’ cmin 11440  β„•0cn0 12468  β„€β‰₯cuz 12818  ...cfz 13480  β†‘cexp 14023  Ξ£csu 15628  Polycply 26037  coeffccoe 26039  degcdgr 26040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-map 8817  df-pm 8818  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-sup 9432  df-inf 9433  df-oi 9500  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-0p 25520  df-ply 26041  df-coe 26043  df-dgr 26044
This theorem is referenced by:  coemul  26105  dgrmul2  26123
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