Theorem coemulhi 26206

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 26185	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2837	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
4		coemulhi.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
5		dgrcl 26185	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
6	4, 5	eqeltrid 2837	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
7		nn0addcl 12427	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
8	3, 6, 7	syl2an 596	. . 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 26204	. . 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 1464	. 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 481	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
14	13	nn0ge0d 12456	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ≤ 𝑁)
15	3	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
16	15	nn0red 12454	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℝ)
17	13	nn0red 12454	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℝ)
18	16, 17	addge01d 11716	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0 ≤ 𝑁 ↔ 𝑀 ≤ (𝑀 + 𝑁)))
19	14, 18	mpbid 232	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ≤ (𝑀 + 𝑁))
20		nn0uz 12780	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	15, 20	eleqtrdi 2843	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (ℤ≥‘0))
22	8	nn0zd 12504	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℤ)
23		elfz5 13423	. . . . . 6 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
24	21, 22, 23	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
25	19, 24	mpbird 257	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (0...(𝑀 + 𝑁)))
26	25	snssd 4762	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → {𝑀} ⊆ (0...(𝑀 + 𝑁)))
27		elsni 4594	. . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀)
28	27	adantl 481	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → 𝑘 = 𝑀)
29		fveq2 6831	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐴‘𝑘) = (𝐴‘𝑀))
30		oveq2 7363	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 + 𝑁) − 𝑀))
31	30	fveq2d 6835	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = (𝐵‘((𝑀 + 𝑁) − 𝑀)))
32	29, 31	oveq12d 7373	. . . . 5 ⊢ (𝑘 = 𝑀 → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))))
33	28, 32	syl 17	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))))
34	16	recnd 11151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℂ)
35	17	recnd 11151	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℂ)
36	34, 35	pncan2d 11485	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
37	36	fveq2d 6835	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘((𝑀 + 𝑁) − 𝑀)) = (𝐵‘𝑁))
38	37	oveq2d 7371	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
39	9	coef3 26184	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
40	39	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
41	40, 15	ffvelcdmd 7027	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴‘𝑀) ∈ ℂ)
42	10	coef3 26184	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
43	42	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
44	43, 13	ffvelcdmd 7027	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘𝑁) ∈ ℂ)
45	41, 44	mulcld 11143	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘𝑁)) ∈ ℂ)
46	38, 45	eqeltrd 2833	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
47	46	adantr 480	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
48	33, 47	eqeltrd 2833	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) ∈ ℂ)
49		simpl 482	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
50		eldifi 4080	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
51		elfznn0 13527	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ ℕ0)
53	9, 1	dgrub 26186	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀)
54	53	3expia 1121	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
55	49, 52, 54	syl2an 596	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
56	55	necon1bd 2947	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
57	56	imp 406	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0)
58	57	oveq1d 7370	. . . . 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 726	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ0⟶ℂ)
60	50	ad2antlr 727	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
61		fznn0sub 13463	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0)
62	60, 61	syl 17	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0)
63	59, 62	ffvelcdmd 7027	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) ∈ ℂ)
64	63	mul02d 11322	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
65	58, 64	eqtrd 2768	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
66	16	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑀 ∈ ℝ)
67	50	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
68	67, 51	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ ℕ0)
69	68	nn0red 12454	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ ℝ)
70	17	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑁 ∈ ℝ)
71	66, 69, 70	leadd1d 11722	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
72	8	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℕ0)
73	72	nn0red 12454	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℝ)
74	73, 69, 70	lesubadd2d 11727	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
75	71, 74	bitr4d 282	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
76	75	notbid 318	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑀 ≤ 𝑘 ↔ ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
77	76	biimpa 476	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
78		simpr 484	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆))
79	50, 61	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0)
80	10, 4	dgrub 26186	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0 ∧ (𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0) → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
81	80	3expia 1121	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
82	78, 79, 81	syl2an 596	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
83	82	necon1bd 2947	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0))
84	83	imp 406	. . . . . . 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 591	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0)
86	85	oveq2d 7371	. . . . 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 726	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝐴:ℕ0⟶ℂ)
88	52	ad2antlr 727	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝑘 ∈ ℕ0)
89	87, 88	ffvelcdmd 7027	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐴‘𝑘) ∈ ℂ)
90	89	mul01d 11323	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · 0) = 0)
91	86, 90	eqtrd 2768	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
92		eldifsni 4743	. . . . . . 7 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ≠ 𝑀)
93	92	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ≠ 𝑀)
94	69, 66	letri3d 11266	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 = 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
95	94	necon3abid 2965	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 ≠ 𝑀 ↔ ¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
96	93, 95	mpbid 232	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘))
97		ianor 983	. . . . 5 ⊢ (¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘) ↔ (¬ 𝑘 ≤ 𝑀 ∨ ¬ 𝑀 ≤ 𝑘))
98	96, 97	sylib 218	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 ∨ ¬ 𝑀 ≤ 𝑘))
99	65, 91, 98	mpjaodan 960	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
100		fzfid 13887	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0...(𝑀 + 𝑁)) ∈ Fin)
101	26, 48, 99, 100	fsumss 15639	. 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 15660	. . . 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 584	. . 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 2768	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
105	12, 101, 104	3eqtr2d 2774	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2929   ∖ cdif 3895  {csn 4577   class class class wbr 5095  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∘_f cof 7617  ℂcc 11015  ℝcr 11016  0cc0 11017   + caddc 11020   · cmul 11022   ≤ cle 11158   − cmin 11355  ℕ₀cn0 12392  ℤcz 12479  ℤ_≥cuz 12742  ...cfz 13414  Σcsu 15600  Polycply 26136  coeffccoe 26138  degcdgr 26139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9542  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094  ax-pre-sup 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-pm 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9337  df-inf 9338  df-oi 9407  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-div 11786  df-nn 12137  df-2 12199  df-3 12200  df-n0 12393  df-z 12480  df-uz 12743  df-rp 12897  df-fz 13415  df-fzo 13562  df-fl 13703  df-seq 13916  df-exp 13976  df-hash 14245  df-cj 15013  df-re 15014  df-im 15015  df-sqrt 15149  df-abs 15150  df-clim 15402  df-rlim 15403  df-sum 15601  df-0p 25618  df-ply 26140  df-coe 26142  df-dgr 26143

This theorem is referenced by:  dgrmul  26223  plymul0or  26235  plydivlem4  26251  vieta1lem2  26266