Theorem coemulhi 26376

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 26355	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	eqeltrid 2873	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
4		coemulhi.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
5		dgrcl 26355	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
6	4, 5	eqeltrid 2873	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
7		nn0addcl 12535	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
8	3, 6, 7	syl2an 607	. . 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 26374	. . 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 1488	. 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 486	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
14	13	nn0ge0d 12564	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ≤ 𝑁)
15	3	adantr 485	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
16	15	nn0red 12562	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℝ)
17	13	nn0red 12562	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℝ)
18	16, 17	addge01d 11798	. . . . . 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 12896	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	15, 20	eleqtrdi 2879	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (ℤ≥‘0))
22	8	nn0zd 12612	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℤ)
23		elfz5 13540	. . . . . 6 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
24	21, 22, 23	syl2anc 595	. . . . 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 4754	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → {𝑀} ⊆ (0...(𝑀 + 𝑁)))
27		elsni 4608	. . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀)
28	27	adantl 486	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → 𝑘 = 𝑀)
29		fveq2 6879	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐴‘𝑘) = (𝐴‘𝑀))
30		oveq2 7416	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 + 𝑁) − 𝑀))
31	30	fveq2d 6883	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = (𝐵‘((𝑀 + 𝑁) − 𝑀)))
32	29, 31	oveq12d 7426	. . . . 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 11233	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℂ)
35	17	recnd 11233	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℂ)
36	34, 35	pncan2d 11567	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
37	36	fveq2d 6883	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘((𝑀 + 𝑁) − 𝑀)) = (𝐵‘𝑁))
38	37	oveq2d 7424	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
39	9	coef3 26354	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
40	39	adantr 485	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
41	40, 15	ffvelcdmd 7078	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴‘𝑀) ∈ ℂ)
42	10	coef3 26354	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
43	42	adantl 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
44	43, 13	ffvelcdmd 7078	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘𝑁) ∈ ℂ)
45	41, 44	mulcld 11225	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘𝑁)) ∈ ℂ)
46	38, 45	eqeltrd 2869	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
47	46	adantr 485	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
48	33, 47	eqeltrd 2869	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) ∈ ℂ)
49		simpl 487	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
50		eldifi 4093	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
51		elfznn0 13644	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
52	50, 51	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ ℕ0)
53	9, 1	dgrub 26356	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀)
54	53	3expia 1137	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
55	49, 52, 54	syl2an 607	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
56	55	necon1bd 2982	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
57	56	imp 411	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0)
58	57	oveq1d 7423	. . . . 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 738	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ0⟶ℂ)
60	50	ad2antlr 739	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
61		fznn0sub 13580	. . . . . . . 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 7078	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) ∈ ℂ)
64	63	mul02d 11404	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
65	58, 64	eqtrd 2804	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
66	16	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑀 ∈ ℝ)
67	50	adantl 486	. . . . . . . . . . . . 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 12562	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ ℝ)
70	17	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑁 ∈ ℝ)
71	66, 69, 70	leadd1d 11804	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
72	8	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℕ0)
73	72	nn0red 12562	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℝ)
74	73, 69, 70	lesubadd2d 11809	. . . . . . . . . 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 481	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
78		simpr 489	. . . . . . . . . 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 26356	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0 ∧ (𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0) → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
81	80	3expia 1137	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
82	78, 79, 81	syl2an 607	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
83	82	necon1bd 2982	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0))
84	83	imp 411	. . . . . . 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 602	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0)
86	85	oveq2d 7424	. . . . 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 738	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝐴:ℕ0⟶ℂ)
88	52	ad2antlr 739	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝑘 ∈ ℕ0)
89	87, 88	ffvelcdmd 7078	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐴‘𝑘) ∈ ℂ)
90	89	mul01d 11405	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · 0) = 0)
91	86, 90	eqtrd 2804	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
92		eldifsni 4759	. . . . . . 7 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ≠ 𝑀)
93	92	adantl 486	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ≠ 𝑀)
94	69, 66	letri3d 11348	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 = 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
95	94	necon3abid 3000	. . . . . 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 14005	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0...(𝑀 + 𝑁)) ∈ Fin)
101	26, 48, 99, 100	fsumss 15772	. 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 15793	. . . 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 595	. . 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 2804	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
105	12, 101, 104	3eqtr2d 2810	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∖ cdif 3910  {csn 4591   class class class wbr 5110  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408   ∘_f cof 7670  ℂcc 11094  ℝcr 11095  0cc0 11096   + caddc 11099   · cmul 11101   ≤ cle 11240   − cmin 11437  ℕ₀cn0 12500  ℤcz 12587  ℤ_≥cuz 12858  ...cfz 13531  Σcsu 15733  Polycply 26306  coeffccoe 26308  degcdgr 26309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-inf2 9606  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-3 12300  df-n0 12501  df-z 12588  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-fl 13821  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734  df-0p 25794  df-ply 26310  df-coe 26312  df-dgr 26313

This theorem is referenced by:  dgrmul  26392  plymul0or  26404  plydivlem4  26422  vieta1lem2  26437