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Theorem coemullem 26562

Description: Lemma for coemul 26564 and dgrmul 26582. (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 26533	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘_f · 𝐺) ∈ (Poly‘ℂ))
2		coeadd.3	. . . . 5 ⊢ 𝑀 = (deg‘𝐹)
3		dgrcl 26545	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
4	2, 3	eqeltrid 2865	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ₀)
5		coeadd.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
6		dgrcl 26545	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
7	5, 6	eqeltrid 2865	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
8		nn0addcl 12634	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 + 𝑁) ∈ ℕ₀)
9	4, 7, 8	syl2an 608	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ₀)
10		fzfid 14109	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (0...𝑛) ∈ Fin)
11		coefv0.1	. . . . . . . . . 10 ⊢ 𝐴 = (coeff‘𝐹)
12	11	coef3 26544	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ₀⟶ℂ)
14	13	adantr 486	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → 𝐴:ℕ₀⟶ℂ)
15		elfznn0 13747	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ₀)
16		ffvelcdm 7079	. . . . . . 7 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
17	14, 15, 16	syl2an 608	. . . . . 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 26544	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ₀⟶ℂ)
20	19	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ₀⟶ℂ)
21	20	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ₀⟶ℂ)
22		fznn0sub 13683	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ₀)
23	22	adantl 487	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 𝑘) ∈ ℕ₀)
24	21, 23	ffvelcdmd 7083	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
25	17, 24	mulcld 11322	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
26	10, 25	fsumcl 15892	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
27	26	fmpttd 7113	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ₀⟶ℂ)
28		oveq2 7426	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29		fvoveq1 7441	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (𝐵‘(𝑛 − 𝑘)) = (𝐵‘(𝑗 − 𝑘)))
30	29	oveq2d 7434	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
31	30	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
32	28, 31	sumeq12dv 15865	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
33		eqid 2761	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))
34		sumex 15848	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) ∈ V
35	32, 33, 34	fvmpt 6991	. . . . . . . . 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 741	. . . . . . . 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 1219	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ 𝑗 ≤ (𝑀 + 𝑁))
38		simp2l 1218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℕ₀)
39	38	nn0red 12661	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℝ)
40		simp3l 1220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ (0...𝑗))
41		elfznn0 13747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ₀)
42	40, 41	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℕ₀)
43	42	nn0red 12661	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℝ)
44	7	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ₀)
45	44	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℕ₀)
46	45	nn0red 12661	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℝ)
47	39, 43, 46	lesubadd2d 11908	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ₀)
49	48	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈ ℕ₀)
50	49	nn0red 12661	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈ ℝ)
51		simp3r 1221	. . . . . . . . . . . . . . . . . . . 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 11923	. . . . . . . . . . . . . . . . . . 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 11331	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
54	50, 46	readdcld 11331	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55		letr 11397	. . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . 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 707	. . . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . . 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 201	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ (𝑗 − 𝑘) ≤ 𝑁)
60		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
61	60	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐺 ∈ (Poly‘𝑆))
62		fznn0sub 13683	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...𝑗) → (𝑗 − 𝑘) ∈ ℕ₀)
63	40, 62	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 − 𝑘) ∈ ℕ₀)
64	18, 5	dgrub 26546	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ₀ ∧ (𝐵‘(𝑗 − 𝑘)) ≠ 0) → (𝑗 − 𝑘) ≤ 𝑁)
65	64	3expia 1139	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ₀) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁))
66	61, 63, 65	syl2anc 596	. . . . . . . . . . . . . . . . 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 2974	. . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐵‘(𝑗 − 𝑘)) = 0)
69	68	oveq2d 7434	. . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐴:ℕ₀⟶ℂ)
71	70, 42	ffvelcdmd 7083	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐴‘𝑘) ∈ ℂ)
72	71	mul01d 11502	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . 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 1139	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
77	76	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆))
78	11, 2	dgrub 26546	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ₀ ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀)
79	78	3expia 1139	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
80	77, 41, 79	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
81	80	necon1bd 2974	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
82	81	imp 412	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0)
83	82	oveq1d 7433	. . . . . . . . . . . 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 743	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ₀⟶ℂ)
85	62	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝑗 − 𝑘) ∈ ℕ₀)
86	84, 85	ffvelcdmd 7083	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘(𝑗 − 𝑘)) ∈ ℂ)
87	86	mul02d 11501	. . . . . . . . . . . 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 2796	. . . . . . . . . . 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 825	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)
90	89	sumeq2dv 15862	. . . . . . . . 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 14108	. . . . . . . . . . 11 ⊢ (0...𝑗) ∈ Fin
92	91	olci 880	. . . . . . . . . 10 ⊢ ((0...𝑗) ⊆ (ℤ_≥‘0) ∨ (0...𝑗) ∈ Fin)
93		sumz 15881	. . . . . . . . . 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 2812	. . . . . . . 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 2796	. . . . . . 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 462	. . . . . 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 2973	. . . . 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 3155	. . . 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 26503	. . . . 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 596	. . . 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 260	. . 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 26547	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
104	103	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “ (ℤ_≥‘(𝑀 + 1))) = {0})
105	18, 5	dgrub2 26547	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
106	105	adantl 487	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “ (ℤ_≥‘(𝑁 + 1))) = {0})
107	11, 2	coeid 26550	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
108	107	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))
109	18, 5	coeid 26550	. . . . . 6 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))
110	109	adantl 487	. . . . 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 26526	. . . 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 13747	. . . . . . . 8 ⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ₀)
113	112, 35	syl 18	. . . . . . 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 7433	. . . . . 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 15858	. . . . 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 5201	. . . 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 2814	. . 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 26539	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘_f · 𝐺)) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))))
119		ffvelcdm 7079	. . . 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 608	. . 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 26555	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘_f · 𝐺)) ≤ (𝑀 + 𝑁))
122	118, 121	jca 521	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ⊆ wss 3899 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 “ cima 5654 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ∘_f cof 7689 Fincfn 8966 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 · cmul 11198 ≤ cle 11337 − cmin 11534 ℕ₀cn0 12599 ℤ_≥cuz 12958 ...cfz 13632 ↑cexp 14197 Σcsu 15846 Polycply 26495 coeffccoe 26497 degcdgr 26498

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-z 12687 df-uz 12959 df-rp 13114 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-rlim 15649 df-sum 15847 df-0p 25984 df-ply 26499 df-coe 26501 df-dgr 26502

This theorem is used by: coemul 26564 dgrmul2 26581

W3C validator