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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‘(𝐹 ∘_f · 𝐺)) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) ∧ (deg‘(𝐹 ∘_f · 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.

Step	Hyp	Ref	Expression
1		plymulcl 26260	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘_f · 𝐺) ∈ (Poly‘ℂ))
2		coeadd.3	. . . . 5 ⊢ 𝑀 = (deg‘𝐹)
3		dgrcl 26272	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
4	2, 3	eqeltrid 2845	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ₀)
5		coeadd.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
6		dgrcl 26272	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
7	5, 6	eqeltrid 2845	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
8		nn0addcl 12561	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 + 𝑁) ∈ ℕ₀)
9	4, 7, 8	syl2an 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)
12	11	coef3 26271	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ₀⟶ℂ)
14	13	adantr 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) → (𝐴‘𝑘) ∈ ℂ)
17	14, 15, 16	syl2an 596	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴‘𝑘) ∈ ℂ)
18		coeadd.2	. . . . . . . . . 10 ⊢ 𝐵 = (coeff‘𝐺)
19	18	coef3 26271	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ₀⟶ℂ)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ₀⟶ℂ)
21	20	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ₀⟶ℂ)
22		fznn0sub 13596	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ₀)
23	22	adantl 481	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 𝑘) ∈ ℕ₀)
24	21, 23	ffvelcdmd 7105	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
25	17, 24	mulcld 11281	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
26	10, 25	fsumcl 15769	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
27	26	fmpttd 7135	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ₀⟶ℂ)
28		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29		fvoveq1 7454	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (𝐵‘(𝑛 − 𝑘)) = (𝐵‘(𝑗 − 𝑘)))
30	29	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
31	30	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
32	28, 31	sumeq12dv 15742	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
33		eqid 2737	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))
34		sumex 15724	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) ∈ V
35	32, 33, 34	fvmpt 7016	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
36	35	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
37		simp2r 1201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ 𝑗 ≤ (𝑀 + 𝑁))
38		simp2l 1200	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℕ₀)
39	38	nn0red 12588	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℝ)
40		simp3l 1202	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ (0...𝑗))
41		elfznn0 13660	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ₀)
42	40, 41	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℕ₀)
43	42	nn0red 12588	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℝ)
44	7	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ₀)
45	44	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℕ₀)
46	45	nn0red 12588	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℝ)
47	39, 43, 46	lesubadd2d 11862	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 − 𝑘) ≤ 𝑁 ↔ 𝑗 ≤ (𝑘 + 𝑁)))
48	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ₀)
49	48	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈ ℕ₀)
50	49	nn0red 12588	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈ ℝ)
51		simp3r 1203	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ≤ 𝑀)
52	43, 50, 46, 51	leadd1dd 11877	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁))
53	43, 46	readdcld 11290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
54	50, 46	readdcld 11290	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (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)))
56	39, 53, 54, 55	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁)))
57	52, 56	mpan2d 694	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 ≤ (𝑘 + 𝑁) → 𝑗 ≤ (𝑀 + 𝑁)))
58	47, 57	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 − 𝑘) ≤ 𝑁 → 𝑗 ≤ (𝑀 + 𝑁)))
59	37, 58	mtod 198	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ (𝑗 − 𝑘) ≤ 𝑁)
60		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
61	60	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐺 ∈ (Poly‘𝑆))
62		fznn0sub 13596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...𝑗) → (𝑗 − 𝑘) ∈ ℕ₀)
63	40, 62	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 − 𝑘) ∈ ℕ₀)
64	18, 5	dgrub 26273	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ₀ ∧ (𝐵‘(𝑗 − 𝑘)) ≠ 0) → (𝑗 − 𝑘) ≤ 𝑁)
65	64	3expia 1122	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ₀) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁))
66	61, 63, 65	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁))
67	66	necon1bd 2958	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (¬ (𝑗 − 𝑘) ≤ 𝑁 → (𝐵‘(𝑗 − 𝑘)) = 0))
68	59, 67	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐵‘(𝑗 − 𝑘)) = 0)
69	68	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = ((𝐴‘𝑘) · 0))
70	13	3ad2ant1 1134	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐴:ℕ₀⟶ℂ)
71	70, 42	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐴‘𝑘) ∈ ℂ)
72	71	mul01d 11460	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · 0) = 0)
73	69, 72	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)
74	73	3expia 1122	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0))
75	74	impl 455	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)
76		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
77	76	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆))
78	11, 2	dgrub 26273	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀)
79	78	3expia 1122	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
80	77, 41, 79	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
81	80	necon1bd 2958	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
82	81	imp 406	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0)
83	82	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = (0 · (𝐵‘(𝑗 − 𝑘))))
84	20	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ₀⟶ℂ)
85	62	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝑗 − 𝑘) ∈ ℕ₀)
86	84, 85	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘(𝑗 − 𝑘)) ∈ ℂ)
87	86	mul02d 11459	. . . . . . . . . . . 12 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘(𝑗 − 𝑘))) = 0)
88	83, 87	eqtrd 2777	. . . . . . . . . . 11 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)
89	75, 88	pm2.61dan 813	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)
90	89	sumeq2dv 15738	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)0)
91		fzfi 14013	. . . . . . . . . . 11 ⊢ (0...𝑗) ∈ Fin
92	91	olci 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)
94	92, 93	ax-mp 5	. . . . . . . . 9 ⊢ Σ𝑘 ∈ (0...𝑗)0 = 0
95	90, 94	eqtrdi 2793	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)
96	36, 95	eqtrd 2777	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = 0)
97	96	expr 456	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ₀) → (¬ 𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = 0))
98	97	necon1ad 2957	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ₀) → (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
99	98	ralrimiva 3146	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))
100		plyco0 26231	. . . . 5 ⊢ (((𝑀 + 𝑁) ∈ ℕ₀ ∧ (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ₀⟶ℂ) → (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “ (ℤ_≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))))
101	9, 27, 100	syl2anc 584	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “ (ℤ_≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑗 ∈ ℕ₀ (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))))
102	99, 101	mpbird 257	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “ (ℤ_≥‘((𝑀 + 𝑁) + 1))) = {0})
103	11, 2	dgrub2 26274	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
104	103	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
105	18, 5	dgrub2 26274	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
106	105	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
107	11, 2	coeid 26277	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
108	107	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
109	18, 5	coeid 26277	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
110	109	adantl 481	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
111	76, 60, 48, 44, 13, 20, 104, 106, 108, 110	plymullem1 26253	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗))))
112		elfznn0 13660	. . . . . . . 8 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ₀)
113	112, 35	syl 17	. . . . . . 7 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
114	113	oveq1d 7446	. . . . . 6 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗)))
115	114	sumeq2i 15734	. . . . 5 ⊢ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗)) = Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗))
116	115	mpteq2i 5247	. . . 4 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗)))
117	111, 116	eqtr4di 2795	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗))))
118	1, 9, 27, 102, 117	coeeq 26266	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘_f · 𝐺)) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))))
119		ffvelcdm 7101	. . . 4 ⊢ (((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ₀⟶ℂ ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ∈ ℂ)
120	27, 112, 119	syl2an 596	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ∈ ℂ)
121	1, 9, 120, 117	dgrle 26282	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘_f · 𝐺)) ≤ (𝑀 + 𝑁))
122	118, 121	jca 511	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘_f · 𝐺)) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) ∧ (deg‘(𝐹 ∘_f · 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 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 ℕ₀cn0 12526 ℤ_≥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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