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Theorem coemullem 26458

Description: Lemma for coemul 26460 and dgrmul 26478. (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 26429	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘_f · 𝐺) ∈ (Poly‘ℂ))
2		coeadd.3	. . . . 5 ⊢ 𝑀 = (deg‘𝐹)
3		dgrcl 26441	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
4	2, 3	eqeltrid 2869	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ₀)
5		coeadd.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
6		dgrcl 26441	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ₀)
7	5, 6	eqeltrid 2869	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ₀)
8		nn0addcl 12554	. . . 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 14027	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (0...𝑛) ∈ Fin)
11		coefv0.1	. . . . . . . . . 10 ⊢ 𝐴 = (coeff‘𝐹)
12	11	coef3 26440	. . . . . . . . 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 13665	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ₀)
16		ffvelcdm 7080	. . . . . . 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 26440	. . . . . . . . 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 13601	. . . . . . . 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 7084	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ)
25	17, 24	mulcld 11244	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
26	10, 25	fsumcl 15807	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ)
27	26	fmpttd 7114	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ₀⟶ℂ)
28		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗))
29		fvoveq1 7442	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (𝐵‘(𝑛 − 𝑘)) = (𝐵‘(𝑗 − 𝑘)))
30	29	oveq2d 7435	. . . . . . . . . . . 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 15780	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))))
33		eqid 2765	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))
34		sumex 15763	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) ∈ V
35	32, 33, 34	fvmpt 6993	. . . . . . . . 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 12581	. . . . . . . . . . . . . . . . . . 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 13665	. . . . . . . . . . . . . . . . . . . . 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 12581	. . . . . . . . . . . . . . . . . . 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 12581	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℝ)
47	39, 43, 46	lesubadd2d 11828	. . . . . . . . . . . . . . . . . 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 12581	. . . . . . . . . . . . . . . . . . . 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 11843	. . . . . . . . . . . . . . . . . . 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 11253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ∈ ℝ)
54	50, 46	readdcld 11253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑀 + 𝑁) ∈ ℝ)
55		letr 11319	. . . . . . . . . . . . . . . . . . . 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 13601	. . . . . . . . . . . . . . . . . . 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 26442	. . . . . . . . . . . . . . . . . . 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 2978	. . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐴‘𝑘) ∈ ℂ)
72	71	mul01d 11424	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 26442	. . . . . . . . . . . . . . . . 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 2978	. . . . . . . . . . . . . 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 7434	. . . . . . . . . . . 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 7084	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ₀ ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘(𝑗 − 𝑘)) ∈ ℂ)
87	86	mul02d 11423	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 15777	. . . . . . . . 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 14026	. . . . . . . . . . 11 ⊢ (0...𝑗) ∈ Fin
92	91	olci 880	. . . . . . . . . 10 ⊢ ((0...𝑗) ⊆ (ℤ_≥‘0) ∨ (0...𝑗) ∈ Fin)
93		sumz 15796	. . . . . . . . . 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 2816	. . . . . . . 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 2800	. . . . . . 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 2977	. . . . 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 3159	. . . 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 26400	. . . . 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 26443	. . . . . 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 26443	. . . . . 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 26446	. . . . . 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 26446	. . . . . 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 26422	. . . 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 13665	. . . . . . . 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 7434	. . . . . 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 15773	. . . . 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 5209	. . . 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 2818	. . 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 26435	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘_f · 𝐺)) = (𝑛 ∈ ℕ₀ ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))))
119		ffvelcdm 7080	. . . 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 26451	. 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 2146 ≠ wne 2960 ∀wral 3081 ⊆ wss 3906 {csn 4591 class class class wbr 5111 ↦ cmpt 5194 “ cima 5666 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∘_f cof 7682 Fincfn 8949 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 ≤ cle 11259 − cmin 11456 ℕ₀cn0 12519 ℤ_≥cuz 12878 ...cfz 13551 ↑cexp 14115 Σcsu 15761 Polycply 26392 coeffccoe 26394 degcdgr 26395

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-n0 12520 df-z 12607 df-uz 12879 df-rp 13033 df-fz 13552 df-fzo 13700 df-fl 13843 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-rlim 15564 df-sum 15762 df-0p 25880 df-ply 26396 df-coe 26398 df-dgr 26399

This theorem is used by: coemul 26460 dgrmul2 26477

W3C validator