Theorem coemulhi 26617

Description: The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
coefv0.1	⊢ 𝐴 = (coeff‘𝐹)
coeadd.2	⊢ 𝐵 = (coeff‘𝐺)
coemulhi.3	⊢ 𝑀 = (deg‘𝐹)
coemulhi.4	⊢ 𝑁 = (deg‘𝐺)

Assertion

Ref	Expression
coemulhi	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁)))

Proof of Theorem coemulhi

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coemulhi.3	. . . . 5 ⊢ 𝑀 = (deg‘𝐹)
2		dgrcl 26596	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2868	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
4		coemulhi.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
5		dgrcl 26596	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
6	4, 5	eqeltrid 2868	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
7		nn0addcl 12679	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
8	3, 6, 7	syl2an 608	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℕ0)
9		coefv0.1	. . . 4 ⊢ 𝐴 = (coeff‘𝐹)
10		coeadd.2	. . . 4 ⊢ 𝐵 = (coeff‘𝐺)
11	9, 10	coemul 26615	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ (𝑀 + 𝑁) ∈ ℕ0) → ((coeff‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))))
12	8, 11	mpd3an3 1492	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))))
13	6	adantl 487	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
14	13	nn0ge0d 12708	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ≤ 𝑁)
15	3	adantr 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
16	15	nn0red 12706	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℝ)
17	13	nn0red 12706	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℝ)
18	16, 17	addge01d 11942	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0 ≤ 𝑁 ↔ 𝑀 ≤ (𝑀 + 𝑁)))
19	14, 18	mpbid 235	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ≤ (𝑀 + 𝑁))
20		nn0uz 13041	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	15, 20	eleqtrdi 2874	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (ℤ≥‘0))
22	8	nn0zd 12756	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℤ)
23		elfz5 13686	. . . . . 6 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
24	21, 22, 23	syl2anc 596	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
25	19, 24	mpbird 260	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (0...(𝑀 + 𝑁)))
26	25	snssd 4747	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → {𝑀} ⊆ (0...(𝑀 + 𝑁)))
27		elsni 4601	. . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀)
28	27	adantl 487	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → 𝑘 = 𝑀)
29		fveq2 6914	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐴‘𝑘) = (𝐴‘𝑀))
30		oveq2 7457	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 + 𝑁) − 𝑀))
31	30	fveq2d 6918	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = (𝐵‘((𝑀 + 𝑁) − 𝑀)))
32	29, 31	oveq12d 7467	. . . . 5 ⊢ (𝑘 = 𝑀 → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))))
33	28, 32	syl 18	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))))
34	16	recnd 11374	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℂ)
35	17	recnd 11374	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℂ)
36	34, 35	pncan2d 11708	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
37	36	fveq2d 6918	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘((𝑀 + 𝑁) − 𝑀)) = (𝐵‘𝑁))
38	37	oveq2d 7465	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
39	9	coef3 26595	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
40	39	adantr 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
41	40, 15	ffvelcdmd 7115	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴‘𝑀) ∈ ℂ)
42	10	coef3 26595	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
43	42	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
44	43, 13	ffvelcdmd 7115	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘𝑁) ∈ ℂ)
45	41, 44	mulcld 11366	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘𝑁)) ∈ ℂ)
46	38, 45	eqeltrd 2864	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
47	46	adantr 486	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
48	33, 47	eqeltrd 2864	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) ∈ ℂ)
49		simpl 488	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
50		eldifi 4070	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
51		elfznn0 13792	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
52	50, 51	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ ℕ0)
53	9, 1	dgrub 26597	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀)
54	53	3expia 1139	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
55	49, 52, 54	syl2an 608	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
56	55	necon1bd 2977	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
57	56	imp 412	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0)
58	57	oveq1d 7464	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = (0 · (𝐵‘((𝑀 + 𝑁) − 𝑘))))
59	43	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ0⟶ℂ)
60	50	ad2antlr 740	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
61		fznn0sub 13728	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0)
62	60, 61	syl 18	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0)
63	59, 62	ffvelcdmd 7115	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) ∈ ℂ)
64	63	mul02d 11545	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
65	58, 64	eqtrd 2798	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
66	16	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑀 ∈ ℝ)
67	50	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
68	67, 51	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ ℕ0)
69	68	nn0red 12706	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ ℝ)
70	17	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑁 ∈ ℝ)
71	66, 69, 70	leadd1d 11948	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
72	8	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℕ0)
73	72	nn0red 12706	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℝ)
74	73, 69, 70	lesubadd2d 11953	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
75	71, 74	bitr4d 285	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
76	75	notbid 321	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑀 ≤ 𝑘 ↔ ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
77	76	biimpa 482	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
78		simpr 490	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
79	50, 61	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0)
80	10, 4	dgrub 26597	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0 ∧ (𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0) → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
81	80	3expia 1139	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
82	78, 79, 81	syl2an 608	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
83	82	necon1bd 2977	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0))
84	83	imp 412	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0)
85	77, 84	syldan 603	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0)
86	85	oveq2d 7465	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑘) · 0))
87	40	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝐴:ℕ0⟶ℂ)
88	52	ad2antlr 740	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝑘 ∈ ℕ0)
89	87, 88	ffvelcdmd 7115	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐴‘𝑘) ∈ ℂ)
90	89	mul01d 11546	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · 0) = 0)
91	86, 90	eqtrd 2798	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
92		eldifsni 4753	. . . . . . 7 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ≠ 𝑀)
93	92	adantl 487	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ≠ 𝑀)
94	69, 66	letri3d 11489	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 = 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
95	94	necon3abid 2995	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 ≠ 𝑀 ↔ ¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
96	93, 95	mpbid 235	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘))
97		ianor 997	. . . . 5 ⊢ (¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘) ↔ (¬ 𝑘 ≤ 𝑀 ∨ ¬ 𝑀 ≤ 𝑘))
98	96, 97	sylib 221	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 ∨ ¬ 𝑀 ≤ 𝑘))
99	65, 91, 98	mpjaodan 973	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
100		fzfid 14154	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0...(𝑀 + 𝑁)) ∈ Fin)
101	26, 48, 99, 100	fsumss 15932	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = Σ𝑘 ∈ (0...(𝑀 + 𝑁))((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))))
102	32	sumsn 15953	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))))
103	15, 46, 102	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))))
104	103, 38	eqtrd 2798	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
105	12, 101, 104	3eqtr2d 2804	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∖ cdif 4048  {csn 4588   class class class wbr 5106  ⟶wf 6560  ‘cfv 6564  (class class class)co 7449   ∘_f cof 7721  ℂcc 11235  ℝcr 11236  0cc0 11237   + caddc 11240   · cmul 11242   ≤ cle 11381   − cmin 11578  ℕ₀cn0 12644  ℤcz 12731  ℤ_≥cuz 13003  ...cfz 13677  Σcsu 15894  Polycply 26546  coeffccoe 26548  degcdgr 26549

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-inf2 9677  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-isom 6573  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-er 8753  df-map 8885  df-pm 8886  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-sup 9469  df-inf 9470  df-oi 9539  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-div 12012  df-nn 12374  df-2 12443  df-3 12444  df-n0 12645  df-z 12732  df-uz 13004  df-rp 13159  df-fz 13678  df-fzo 13827  df-fl 13970  df-seq 14183  df-exp 14243  df-hash 14514  df-cj 15307  df-re 15308  df-im 15309  df-sqrt 15443  df-abs 15444  df-clim 15696  df-rlim 15697  df-sum 15895  df-0p 26035  df-ply 26550  df-coe 26552  df-dgr 26553

This theorem is used by:  dgrmul  26633  plymul0or  26643  plydivlem4  26661  vieta1lem2  26678