Theorem coemulc 26188

Description: The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)

Assertion

Ref	Expression
coemulc	⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = ((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹)))

Proof of Theorem coemulc

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3957	. . . . 5 ⊢ ℂ ⊆ ℂ
2		plyconst 26139	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
3	1, 2	mpan 690	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
4		plyssc 26133	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
5	4	sseli 3930	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
6		plymulcl 26154	. . . 4 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ)) → ((ℂ × {𝐴}) ∘f · 𝐹) ∈ (Poly‘ℂ))
7	3, 5, 6	syl2an 596	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ × {𝐴}) ∘f · 𝐹) ∈ (Poly‘ℂ))
8		eqid 2731	. . . 4 ⊢ (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = (coeff‘((ℂ × {𝐴}) ∘f · 𝐹))
9	8	coef3 26165	. . 3 ⊢ (((ℂ × {𝐴}) ∘f · 𝐹) ∈ (Poly‘ℂ) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)):ℕ0⟶ℂ)
10		ffn 6651	. . 3 ⊢ ((coeff‘((ℂ × {𝐴}) ∘f · 𝐹)):ℕ0⟶ℂ → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) Fn ℕ0)
11	7, 9, 10	3syl 18	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) Fn ℕ0)
12		fconstg 6710	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
13	12	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
14	13	ffnd 6652	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}) Fn ℕ0)
15		eqid 2731	. . . . . 6 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
16	15	coef3 26165	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
17	16	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
18	17	ffnd 6652	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn ℕ0)
19		nn0ex 12387	. . . 4 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21		inidm 4177	. . 3 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
22	14, 18, 20, 20, 21	offn 7623	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹)) Fn ℕ0)
23	3	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
24		eqid 2731	. . . . . . 7 ⊢ (coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴}))
25	24	coefv0 26181	. . . . . 6 ⊢ ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
26	23, 25	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
27		simpll 766	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ ℂ)
28		0cn 11104	. . . . . 6 ⊢ 0 ∈ ℂ
29		fvconst2g 7136	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
30	27, 28, 29	sylancl 586	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = 𝐴)
31	26, 30	eqtr3d 2768	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(ℂ × {𝐴}))‘0) = 𝐴)
32		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
33	32	nn0cnd 12444	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
34	33	subid1d 11461	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝑛 − 0) = 𝑛)
35	34	fveq2d 6826	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛))
36	31, 35	oveq12d 7364	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
37	5	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
38	24, 15	coemul 26185	. . . . 5 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))))
39	23, 37, 32, 38	syl3anc 1373	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))))
40		nn0uz 12774	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
41	32, 40	eleqtrdi 2841	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ≥‘0))
42		fzss2 13464	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘0) → (0...0) ⊆ (0...𝑛))
43	41, 42	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...0) ⊆ (0...𝑛))
44		elfz1eq 13435	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
45	44	adantl 481	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
46		fveq2 6822	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0))
47		oveq2 7354	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
48	47	fveq2d 6826	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛 − 𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0)))
49	46, 48	oveq12d 7364	. . . . . . 7 ⊢ (𝑘 = 0 → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
50	45, 49	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
51	17	ffvelcdmda 7017	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) ∈ ℂ)
52	27, 51	mulcld 11132	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ)
53	36, 52	eqeltrd 2831	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
54	53	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
55	50, 54	eqeltrd 2831	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) ∈ ℂ)
56		eldifn 4082	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → ¬ 𝑘 ∈ (0...0))
57	56	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ¬ 𝑘 ∈ (0...0))
58		eldifi 4081	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛))
59		elfznn0 13520	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
60	58, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ0)
61		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴}))
62	24, 61	dgrub 26167	. . . . . . . . . . . . 13 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(ℂ × {𝐴})))
63	62	3expia 1121	. . . . . . . . . . . 12 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
64	23, 60, 63	syl2an 596	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
65		0dgr 26178	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
66	65	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (deg‘(ℂ × {𝐴})) = 0)
67	66	breq2d 5103	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 ≤ 0))
68	60	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → 𝑘 ∈ ℕ0)
69		nn0le0eq0 12409	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (𝑘 ≤ 0 ↔ 𝑘 = 0))
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ 0 ↔ 𝑘 = 0))
71	67, 70	bitrd 279	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 = 0))
72	64, 71	sylibd 239	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 = 0))
73		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → 𝑘 = 0)
74		0z 12479	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
75		elfz3 13434	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
76	74, 75	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...0)
77	73, 76	eqeltrdi 2839	. . . . . . . . . 10 ⊢ (𝑘 = 0 → 𝑘 ∈ (0...0))
78	72, 77	syl6 35	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ∈ (0...0)))
79	78	necon1bd 2946	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (¬ 𝑘 ∈ (0...0) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0))
80	57, 79	mpd 15	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0)
81	80	oveq1d 7361	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (0 · ((coeff‘𝐹)‘(𝑛 − 𝑘))))
82	17	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹):ℕ0⟶ℂ)
83		fznn0sub 13456	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
84	58, 83	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛 − 𝑘) ∈ ℕ0)
85		ffvelcdm 7014	. . . . . . . 8 ⊢ (((coeff‘𝐹):ℕ0⟶ℂ ∧ (𝑛 − 𝑘) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ)
86	82, 84, 85	syl2an 596	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ)
87	86	mul02d 11311	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
88	81, 87	eqtrd 2766	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
89		fzfid 13880	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
90	43, 55, 88, 89	fsumss 15632	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))))
91	49	fsum1 15654	. . . . 5 ⊢ ((0 ∈ ℤ ∧ (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
92	74, 53, 91	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → Σ𝑘 ∈ (0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
93	39, 90, 92	3eqtr2d 2772	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
94		simpl 482	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐴 ∈ ℂ)
95		eqidd 2732	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛))
96	20, 94, 18, 95	ofc1 7638	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
97	36, 93, 96	3eqtr4d 2776	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = (((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹))‘𝑛))
98	11, 22, 97	eqfnfvd 6967	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = ((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436   ∖ cdif 3899   ⊆ wss 3902  {csn 4576   class class class wbr 5091   × cxp 5614   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608  ℂcc 11004  0cc0 11006   · cmul 11011   ≤ cle 11147   − cmin 11344  ℕ₀cn0 12381  ℤcz 12468  ℤ_≥cuz 12732  ...cfz 13407  Σcsu 15593  Polycply 26117  coeffccoe 26119  degcdgr 26120

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-rp 12891  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-0p 25599  df-ply 26121  df-coe 26123  df-dgr 26124

This theorem is referenced by:  coe0  26189  coesub  26190  mpaaeu  43189