Theorem coemulc 24011

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 3624	. . . . 5 ⊢ ℂ ⊆ ℂ
2		plyconst 23962	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
3	1, 2	mpan 706	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
4		plyssc 23956	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
5	4	sseli 3599	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
6		plymulcl 23977	. . . 4 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ)) → ((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ))
7	3, 5, 6	syl2an 494	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ))
8		eqid 2622	. . . 4 ⊢ (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))
9	8	coef3 23988	. . 3 ⊢ (((ℂ × {𝐴}) ∘𝑓 · 𝐹) ∈ (Poly‘ℂ) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)):ℕ0⟶ℂ)
10		ffn 6045	. . 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 6092	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
13	12	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}):ℕ0⟶{𝐴})
14		ffn 6045	. . . 4 ⊢ ((ℕ0 × {𝐴}):ℕ0⟶{𝐴} → (ℕ0 × {𝐴}) Fn ℕ0)
15	13, 14	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0 × {𝐴}) Fn ℕ0)
16		eqid 2622	. . . . . 6 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
17	16	coef3 23988	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
18	17	adantl 482	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
19		ffn 6045	. . . 4 ⊢ ((coeff‘𝐹):ℕ0⟶ℂ → (coeff‘𝐹) Fn ℕ0)
20	18, 19	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn ℕ0)
21		nn0ex 11298	. . . 4 ⊢ ℕ0 ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ0 ∈ V)
23		inidm 3822	. . 3 ⊢ (ℕ0 ∩ ℕ0) = ℕ0
24	15, 20, 22, 22, 23	offn 6908	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹)) Fn ℕ0)
25	3	ad2antrr 762	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
26		eqid 2622	. . . . . . 7 ⊢ (coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴}))
27	26	coefv0 24004	. . . . . 6 ⊢ ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
28	25, 27	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ × {𝐴}))‘0))
29		simpll 790	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈ ℂ)
30		0cn 10032	. . . . . 6 ⊢ 0 ∈ ℂ
31		fvconst2g 6467	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
32	29, 30, 31	sylancl 694	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ × {𝐴})‘0) = 𝐴)
33	28, 32	eqtr3d 2658	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(ℂ × {𝐴}))‘0) = 𝐴)
34		simpr 477	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
35	34	nn0cnd 11353	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
36	35	subid1d 10381	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝑛 − 0) = 𝑛)
37	36	fveq2d 6195	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛))
38	33, 37	oveq12d 6668	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
39	5	ad2antlr 763	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℂ))
40	26, 16	coemul 24008	. . . . 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))))
41	25, 39, 34, 40	syl3anc 1326	. . . 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))))
42		nn0uz 11722	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
43	34, 42	syl6eleq 2711	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ (ℤ≥‘0))
44		fzss2 12381	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘0) → (0...0) ⊆ (0...𝑛))
45	43, 44	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...0) ⊆ (0...𝑛))
46		elfz1eq 12352	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
47	46	adantl 482	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
48		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0))
49		oveq2 6658	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
50	49	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛 − 𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0)))
51	48, 50	oveq12d 6668	. . . . . . 7 ⊢ (𝑘 = 0 → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
52	47, 51	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))))
53	18	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) ∈ ℂ)
54	29, 53	mulcld 10060	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ)
55	38, 54	eqeltrd 2701	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
56	55	adantr 481	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
57	52, 56	eqeltrd 2701	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) ∈ ℂ)
58		eldifn 3733	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → ¬ 𝑘 ∈ (0...0))
59	58	adantl 482	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ¬ 𝑘 ∈ (0...0))
60		eldifi 3732	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛))
61		elfznn0 12433	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0)
62	60, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ0)
63		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴}))
64	26, 63	dgrub 23990	. . . . . . . . . . . . 13 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(ℂ × {𝐴})))
65	64	3expia 1267	. . . . . . . . . . . 12 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
66	25, 62, 65	syl2an 494	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
67		0dgr 24001	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
68	67	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (deg‘(ℂ × {𝐴})) = 0)
69	68	breq2d 4665	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 ≤ 0))
70	62	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → 𝑘 ∈ ℕ0)
71		nn0le0eq0 11321	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (𝑘 ≤ 0 ↔ 𝑘 = 0))
72	70, 71	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ 0 ↔ 𝑘 = 0))
73	69, 72	bitrd 268	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 = 0))
74	66, 73	sylibd 229	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 = 0))
75		id 22	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → 𝑘 = 0)
76		0z 11388	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
77		elfz3 12351	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
78	76, 77	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...0)
79	75, 78	syl6eqel 2709	. . . . . . . . . 10 ⊢ (𝑘 = 0 → 𝑘 ∈ (0...0))
80	74, 79	syl6 35	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ∈ (0...0)))
81	80	necon1bd 2812	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (¬ 𝑘 ∈ (0...0) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0))
82	59, 81	mpd 15	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘(ℂ × {𝐴}))‘𝑘) = 0)
83	82	oveq1d 6665	. . . . . 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))))
84	18	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (coeff‘𝐹):ℕ0⟶ℂ)
85		fznn0sub 12373	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ0)
86	60, 85	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛 − 𝑘) ∈ ℕ0)
87		ffvelrn 6357	. . . . . . . 8 ⊢ (((coeff‘𝐹):ℕ0⟶ℂ ∧ (𝑛 − 𝑘) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ)
88	84, 86, 87	syl2an 494	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ)
89	88	mul02d 10234	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
90	83, 89	eqtrd 2656	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
91		fzfid 12772	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...𝑛) ∈ Fin)
92	45, 57, 90, 91	fsumss 14456	. . . 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))))
93	51	fsum1 14476	. . . . 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))))
94	76, 55, 93	sylancr 695	. . . 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))))
95	41, 92, 94	3eqtr2d 2662	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
96		simpl 473	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐴 ∈ ℂ)
97		eqidd 2623	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛))
98	22, 96, 20, 97	ofc1 6920	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
99	38, 95, 98	3eqtr4d 2666	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹))‘𝑛) = (((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹))‘𝑛))
100	11, 24, 99	eqfnfvd 6314	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = ((ℕ0 × {𝐴}) ∘𝑓 · (coeff‘𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177   class class class wbr 4653   × cxp 5112   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ℂcc 9934  0cc0 9936   · 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:  coe0  24012  coesub  24013  mpaaeu  37720