Theorem coemulhi 24010

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 23989	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
3	1, 2	syl5eqel 2705	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈ ℕ0)
4		coemulhi.4	. . . . 5 ⊢ 𝑁 = (deg‘𝐺)
5		dgrcl 23989	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
6	4, 5	syl5eqel 2705	. . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0)
7		nn0addcl 11328	. . . 4 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ ℕ0)
8	3, 6, 7	syl2an 494	. . 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 24008	. . 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 1425	. 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 482	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℕ0)
14	13	nn0ge0d 11354	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 0 ≤ 𝑁)
15	3	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℕ0)
16	15	nn0red 11352	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℝ)
17	13	nn0red 11352	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℝ)
18	16, 17	addge01d 10615	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0 ≤ 𝑁 ↔ 𝑀 ≤ (𝑀 + 𝑁)))
19	14, 18	mpbid 222	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ≤ (𝑀 + 𝑁))
20		nn0uz 11722	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	15, 20	syl6eleq 2711	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (ℤ≥‘0))
22	8	nn0zd 11480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈ ℤ)
23		elfz5 12334	. . . . . 6 ⊢ ((𝑀 ∈ (ℤ≥‘0) ∧ (𝑀 + 𝑁) ∈ ℤ) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
24	21, 22, 23	syl2anc 693	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 ∈ (0...(𝑀 + 𝑁)) ↔ 𝑀 ≤ (𝑀 + 𝑁)))
25	19, 24	mpbird 247	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ (0...(𝑀 + 𝑁)))
26	25	snssd 4340	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → {𝑀} ⊆ (0...(𝑀 + 𝑁)))
27		elsni 4194	. . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀)
28	27	adantl 482	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → 𝑘 = 𝑀)
29		fveq2 6191	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐴‘𝑘) = (𝐴‘𝑀))
30		oveq2 6658	. . . . . . 7 ⊢ (𝑘 = 𝑀 → ((𝑀 + 𝑁) − 𝑘) = ((𝑀 + 𝑁) − 𝑀))
31	30	fveq2d 6195	. . . . . 6 ⊢ (𝑘 = 𝑀 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = (𝐵‘((𝑀 + 𝑁) − 𝑀)))
32	29, 31	oveq12d 6668	. . . . 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 10068	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈ ℂ)
35	17	recnd 10068	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈ ℂ)
36	34, 35	pncan2d 10394	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑀 + 𝑁) − 𝑀) = 𝑁)
37	36	fveq2d 6195	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘((𝑀 + 𝑁) − 𝑀)) = (𝐵‘𝑁))
38	37	oveq2d 6666	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
39	9	coef3 23988	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
40	39	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ)
41	40, 15	ffvelrnd 6360	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴‘𝑀) ∈ ℂ)
42	10	coef3 23988	. . . . . . . . 9 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ)
43	42	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ)
44	43, 13	ffvelrnd 6360	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵‘𝑁) ∈ ℂ)
45	41, 44	mulcld 10060	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘𝑁)) ∈ ℂ)
46	38, 45	eqeltrd 2701	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
47	46	adantr 481	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑀) · (𝐵‘((𝑀 + 𝑁) − 𝑀))) ∈ ℂ)
48	33, 47	eqeltrd 2701	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ {𝑀}) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) ∈ ℂ)
49		simpl 473	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆))
50		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
51		elfznn0 12433	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...(𝑀 + 𝑁)) → 𝑘 ∈ ℕ0)
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ∈ ℕ0)
53	9, 1	dgrub 23990	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀)
54	53	3expia 1267	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
55	49, 52, 54	syl2an 494	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀))
56	55	necon1bd 2812	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0))
57	56	imp 445	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0)
58	57	oveq1d 6665	. . . . 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 762	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ0⟶ℂ)
60	50	ad2antlr 763	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → 𝑘 ∈ (0...(𝑀 + 𝑁)))
61		fznn0sub 12373	. . . . . . . 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	ffvelrnd 6360	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) ∈ ℂ)
64	63	mul02d 10234	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
65	58, 64	eqtrd 2656	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
66	16	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑀 ∈ ℝ)
67	50	adantl 482	. . . . . . . . . . . . 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 11352	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ∈ ℝ)
70	17	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑁 ∈ ℝ)
71	66, 69, 70	leadd1d 10621	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
72	8	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℕ0)
73	72	nn0red 11352	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 + 𝑁) ∈ ℝ)
74	73, 69, 70	lesubadd2d 10626	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 ↔ (𝑀 + 𝑁) ≤ (𝑘 + 𝑁)))
75	71, 74	bitr4d 271	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑀 ≤ 𝑘 ↔ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
76	75	notbid 308	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑀 ≤ 𝑘 ↔ ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
77	76	biimpa 501	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
78		simpr 477	. . . . . . . . . 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 23990	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0 ∧ (𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0) → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁)
81	80	3expia 1267	. . . . . . . . . 10 ⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ ((𝑀 + 𝑁) − 𝑘) ∈ ℕ0) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
82	78, 79, 81	syl2an 494	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐵‘((𝑀 + 𝑁) − 𝑘)) ≠ 0 → ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁))
83	82	necon1bd 2812	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ ((𝑀 + 𝑁) − 𝑘) ≤ 𝑁 → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0))
84	83	imp 445	. . . . . . 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 487	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐵‘((𝑀 + 𝑁) − 𝑘)) = 0)
86	85	oveq2d 6666	. . . . 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 762	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝐴:ℕ0⟶ℂ)
88	52	ad2antlr 763	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → 𝑘 ∈ ℕ0)
89	87, 88	ffvelrnd 6360	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → (𝐴‘𝑘) ∈ ℂ)
90	89	mul01d 10235	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · 0) = 0)
91	86, 90	eqtrd 2656	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) ∧ ¬ 𝑀 ≤ 𝑘) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
92		eldifsni 4320	. . . . . . 7 ⊢ (𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀}) → 𝑘 ≠ 𝑀)
93	92	adantl 482	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → 𝑘 ≠ 𝑀)
94	69, 66	letri3d 10179	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 = 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
95	94	necon3abid 2830	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (𝑘 ≠ 𝑀 ↔ ¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘)))
96	93, 95	mpbid 222	. . . . 5 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘))
97		ianor 509	. . . . 5 ⊢ (¬ (𝑘 ≤ 𝑀 ∧ 𝑀 ≤ 𝑘) ↔ (¬ 𝑘 ≤ 𝑀 ∨ ¬ 𝑀 ≤ 𝑘))
98	96, 97	sylib 208	. . . 4 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → (¬ 𝑘 ≤ 𝑀 ∨ ¬ 𝑀 ≤ 𝑘))
99	65, 91, 98	mpjaodan 827	. . 3 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑘 ∈ ((0...(𝑀 + 𝑁)) ∖ {𝑀})) → ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = 0)
100		fzfid 12772	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (0...(𝑀 + 𝑁)) ∈ Fin)
101	26, 48, 99, 100	fsumss 14456	. 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 14475	. . . 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 693	. . 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 2656	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → Σ𝑘 ∈ {𝑀} ((𝐴‘𝑘) · (𝐵‘((𝑀 + 𝑁) − 𝑘))) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
105	12, 101, 104	3eqtr2d 2662	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘𝑓 · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571  {csn 4177   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939   · cmul 9941   ≤ cle 10075   − cmin 10266  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  Σcsu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  dgrmul  24026  plymul0or  24036  plydivlem4  24051  vieta1lem2  24066