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Theorem coemulc 25769

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 4005	. . . . 5 ⊢ ℂ ⊆ ℂ
2		plyconst 25720	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
3	1, 2	mpan 689	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
4		plyssc 25714	. . . . 5 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
5	4	sseli 3979	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
6		plymulcl 25735	. . . 4 ⊢ (((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ)) → ((ℂ × {𝐴}) ∘_f · 𝐹) ∈ (Poly‘ℂ))
7	3, 5, 6	syl2an 597	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ × {𝐴}) ∘_f · 𝐹) ∈ (Poly‘ℂ))
8		eqid 2733	. . . 4 ⊢ (coeff‘((ℂ × {𝐴}) ∘_f · 𝐹)) = (coeff‘((ℂ × {𝐴}) ∘_f · 𝐹))
9	8	coef3 25746	. . 3 ⊢ (((ℂ × {𝐴}) ∘_f · 𝐹) ∈ (Poly‘ℂ) → (coeff‘((ℂ × {𝐴}) ∘_f · 𝐹)):ℕ₀⟶ℂ)
10		ffn 6718	. . 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 6779	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℕ₀ × {𝐴}):ℕ₀⟶{𝐴})
13	12	adantr 482	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ₀ × {𝐴}):ℕ₀⟶{𝐴})
14	13	ffnd 6719	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ₀ × {𝐴}) Fn ℕ₀)
15		eqid 2733	. . . . . 6 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
16	15	coef3 25746	. . . . 5 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶ℂ)
17	16	adantl 483	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ₀⟶ℂ)
18	17	ffnd 6719	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn ℕ₀)
19		nn0ex 12478	. . . 4 ⊢ ℕ₀ ∈ V
20	19	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ₀ ∈ V)
21		inidm 4219	. . 3 ⊢ (ℕ₀ ∩ ℕ₀) = ℕ₀
22	14, 18, 20, 20, 21	offn 7683	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℕ₀ × {𝐴}) ∘_f · (coeff‘𝐹)) Fn ℕ₀)
23	3	ad2antrr 725	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (ℂ × {𝐴}) ∈ (Poly‘ℂ))
24		eqid 2733	. . . . . . 7 ⊢ (coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴}))
25	24	coefv0 25762	. . . . . 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 11206	. . . . . 6 ⊢ 0 ∈ ℂ
29		fvconst2g 7203	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((ℂ × {𝐴})‘0) = 𝐴)
30	27, 28, 29	sylancl 587	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((ℂ × {𝐴})‘0) = 𝐴)
31	26, 30	eqtr3d 2775	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((coeff‘(ℂ × {𝐴}))‘0) = 𝐴)
32		simpr 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
33	32	nn0cnd 12534	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℂ)
34	33	subid1d 11560	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (𝑛 − 0) = 𝑛)
35	34	fveq2d 6896	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛))
36	31, 35	oveq12d 7427	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
37	5	ad2antlr 726	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → 𝐹 ∈ (Poly‘ℂ))
38	24, 15	coemul 25766	. . . . 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 1372	. . . 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 12864	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
41	32, 40	eleqtrdi 2844	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ (ℤ_≥‘0))
42		fzss2 13541	. . . . . 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 13512	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
45	44	adantl 483	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0)
46		fveq2 6892	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0))
47		oveq2 7417	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0))
48	47	fveq2d 6896	. . . . . . . 8 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛 − 𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0)))
49	46, 48	oveq12d 7427	. . . . . . 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 7087	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((coeff‘𝐹)‘𝑛) ∈ ℂ)
52	27, 51	mulcld 11234	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ)
53	36, 52	eqeltrd 2834	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
54	53	adantr 482	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ)
55	50, 54	eqeltrd 2834	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...0)) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) ∈ ℂ)
56		eldifn 4128	. . . . . . . . 9 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → ¬ 𝑘 ∈ (0...0))
57	56	adantl 483	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ¬ 𝑘 ∈ (0...0))
58		eldifi 4127	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛))
59		elfznn0 13594	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ₀)
60	58, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ₀)
61		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴}))
62	24, 61	dgrub 25748	. . . . . . . . . . . . 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 597	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴}))))
65		0dgr 25759	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
66	65	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (deg‘(ℂ × {𝐴})) = 0)
67	66	breq2d 5161	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ × {𝐴})) ↔ 𝑘 ≤ 0))
68	60	adantl 483	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → 𝑘 ∈ ℕ₀)
69		nn0le0eq0 12500	. . . . . . . . . . . . 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 238	. . . . . . . . . 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 12569	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
75		elfz3 13511	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → 0 ∈ (0...0))
76	74, 75	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...0)
77	73, 76	eqeltrdi 2842	. . . . . . . . . 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 2959	. . . . . . . 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 7424	. . . . . 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 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (coeff‘𝐹):ℕ₀⟶ℂ)
83		fznn0sub 13533	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈ ℕ₀)
84	58, 83	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛 − 𝑘) ∈ ℕ₀)
85		ffvelcdm 7084	. . . . . . . 8 ⊢ (((coeff‘𝐹):ℕ₀⟶ℂ ∧ (𝑛 − 𝑘) ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ)
86	82, 84, 85	syl2an 597	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ)
87	86	mul02d 11412	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
88	81, 87	eqtrd 2773	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0)
89		fzfid 13938	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (0...𝑛) ∈ Fin)
90	43, 55, 88, 89	fsumss 15671	. . . 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 15693	. . . . 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 588	. . . 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 2779	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((coeff‘((ℂ × {𝐴}) ∘_f · 𝐹))‘𝑛) = (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))))
94		simpl 484	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐴 ∈ ℂ)
95		eqidd 2734	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → ((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛))
96	20, 94, 18, 95	ofc1 7696	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ₀) → (((ℕ₀ × {𝐴}) ∘_f · (coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛)))
97	36, 93, 96	3eqtr4d 2783	. 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 7036	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘_f · 𝐹)) = ((ℕ₀ × {𝐴}) ∘_f · (coeff‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∖ cdif 3946 ⊆ wss 3949 {csn 4629 class class class wbr 5149 × cxp 5675 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∘_f cof 7668 ℂcc 11108 0cc0 11110 · cmul 11115 ≤ cle 11249 − cmin 11444 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 Σcsu 15632 Polycply 25698 coeffccoe 25700 degcdgr 25701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-0p 25187 df-ply 25702 df-coe 25704 df-dgr 25705

This theorem is referenced by: coe0 25770 coesub 25771 mpaaeu 41892

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