Theorem coemulc 26294

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 4006	. . . . 5 ⊢ ℂ ⊆ ℂ
2		plyconst 26245	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
3	1, 2	mpan 690	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
4		plyssc 26239	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
5	4	sseli 3979	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
6		plymulcl 26260	. . . 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 2737	. . . 4 ⊢ (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = (coeff‘((ℂ × {𝐴}) ∘f · 𝐹))
9	8	coef3 26271	. . 3 ⊢ (((ℂ × {𝐴}) ∘f · 𝐹) ∈ (Poly‘ℂ) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)):ℕ0⟶ℂ)
10		ffn 6736	. . 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 6795	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
13	12	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
14	13	ffnd 6737	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}) Fn ℕ0)
15		eqid 2737	. . . . . 6 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
16	15	coef3 26271	. . . . 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 6737	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn ℕ0)
19		nn0ex 12532	. . . 4 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
21		inidm 4227	. . 3 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
22	14, 18, 20, 20, 21	offn 7710	. 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 2737	. . . . . . 7 ⊢ (coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴}))
25	24	coefv0 26287	. . . . . 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 767	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ ℂ)
28		0cn 11253	. . . . . 6 ⊢ 0 ∈ ℂ
29		fvconst2g 7222	. . . . . 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 2779	. . . 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 12589	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
34	33	subid1d 11609	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝑛 − 0) = 𝑛)
35	34	fveq2d 6910	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛))
36	31, 35	oveq12d 7449	. . 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 26291	. . . . 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 12920	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
41	32, 40	eleqtrdi 2851	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ≥‘0))
42		fzss2 13604	. . . . . 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 13575	. . . . . . . 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 6906	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0))
47		oveq2 7439	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
48	47	fveq2d 6910	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛 − 𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0)))
49	46, 48	oveq12d 7449	. . . . . . 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 7104	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) ∈ ℂ)
52	27, 51	mulcld 11281	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ)
53	36, 52	eqeltrd 2841	. . . . . . 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 2841	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) ∈ ℂ)
56		eldifn 4132	. . . . . . . . 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 4131	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛))
59		elfznn0 13660	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
60	58, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ0)
61		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴}))
62	24, 61	dgrub 26273	. . . . . . . . . . . . 13 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(ℂ × {𝐴})))
63	62	3expia 1122	. . . . . . . . . . . 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 26284	. . . . . . . . . . . . . 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 5155	. . . . . . . . . . . 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 12554	. . . . . . . . . . . . 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 12624	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
75		elfz3 13574	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
76	74, 75	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...0)
77	73, 76	eqeltrdi 2849	. . . . . . . . . 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 2958	. . . . . . . 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 7446	. . . . . 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 13596	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
84	58, 83	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛 − 𝑘) ∈ ℕ0)
85		ffvelcdm 7101	. . . . . . . 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 11459	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
88	81, 87	eqtrd 2777	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
89		fzfid 14014	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
90	43, 55, 88, 89	fsumss 15761	. . . 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 15783	. . . . 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 2783	. . 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 2738	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛))
96	20, 94, 18, 95	ofc1 7725	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
97	36, 93, 96	3eqtr4d 2787	. 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 7054	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 1540   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  {csn 4626   class class class wbr 5143   × cxp 5683   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695  ℂcc 11153  0cc0 11155   · cmul 11160   ≤ cle 11296   − cmin 11492  ℕ₀cn0 12526  ℤcz 12613  ℤ_≥cuz 12878  ...cfz 13547  Σcsu 15722  Polycply 26223  coeffccoe 26225  degcdgr 26226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-0p 25705  df-ply 26227  df-coe 26229  df-dgr 26230

This theorem is referenced by:  coe0  26295  coesub  26296  mpaaeu  43162