Theorem aareccl 26262

Description: The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)

Assertion

Ref	Expression
aareccl	⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Proof of Theorem aareccl

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

Step	Hyp	Ref	Expression
1		elaa 26252	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
2	1	simprbi 496	. . 3 ⊢ (𝐴 ∈ 𝔸 → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
3	2	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
4		aacn 26253	. . . . 5 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
5		reccl 11790	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
6	4, 5	sylan 580	. . . 4 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
7	6	adantr 480	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ ℂ)
8		zsscn 12483	. . . . . . 7 ⊢ ℤ ⊆ ℂ
9	8	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ℤ ⊆ ℂ)
10		simprl 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
11		eldifsn 4737	. . . . . . . . 9 ⊢ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ (𝑓 ∈ (Poly‘ℤ) ∧ 𝑓 ≠ 0𝑝))
12	10, 11	sylib 218	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ (Poly‘ℤ) ∧ 𝑓 ≠ 0𝑝))
13	12	simpld 494	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ (Poly‘ℤ))
14		dgrcl 26166	. . . . . . 7 ⊢ (𝑓 ∈ (Poly‘ℤ) → (deg‘𝑓) ∈ ℕ0)
15	13, 14	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℕ0)
16	13	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑓 ∈ (Poly‘ℤ))
17		0z 12486	. . . . . . . 8 ⊢ 0 ∈ ℤ
18		eqid 2733	. . . . . . . . 9 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
19	18	coef2 26164	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑓):ℕ0⟶ℤ)
20	16, 17, 19	sylancl 586	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (coeff‘𝑓):ℕ0⟶ℤ)
21		fznn0sub 13458	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
22	21	adantl 481	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
23	20, 22	ffvelcdmd 7024	. . . . . 6 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) ∈ ℤ)
24	9, 15, 23	elplyd 26135	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ∈ (Poly‘ℤ))
25		0cn 11111	. . . . . 6 ⊢ 0 ∈ ℂ
26		eqid 2733	. . . . . . . . . 10 ⊢ (coeff‘(𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))) = (coeff‘(𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))))
27	26	coefv0 26181	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ∈ (Poly‘ℤ) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘0) = ((coeff‘(𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))))‘0))
28	24, 27	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘0) = ((coeff‘(𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))))‘0))
29	23	zcnd 12584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) ∈ ℂ)
30		eqidd 2734	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))))
31	24, 15, 29, 30	coeeq2 26175	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (coeff‘(𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0)))
32	31	fveq1d 6830	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((coeff‘(𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))))‘0) = ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0))‘0))
33		0nn0 12403	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
34		breq1 5096	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
35		oveq2 7360	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((deg‘𝑓) − 𝑘) = ((deg‘𝑓) − 0))
36	35	fveq2d 6832	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) = ((coeff‘𝑓)‘((deg‘𝑓) − 0)))
37	34, 36	ifbieq1d 4499	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0) = if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0))
38		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0))
39		fvex 6841	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑓)‘((deg‘𝑓) − 0)) ∈ V
40		c0ex 11113	. . . . . . . . . . . 12 ⊢ 0 ∈ V
41	39, 40	ifex 4525	. . . . . . . . . . 11 ⊢ if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0) ∈ V
42	37, 38, 41	fvmpt 6935	. . . . . . . . . 10 ⊢ (0 ∈ ℕ0 → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0))‘0) = if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0))
43	33, 42	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0))‘0) = if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0)
44	15	nn0ge0d 12452	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 0 ≤ (deg‘𝑓))
45	44	iftrued 4482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0) = ((coeff‘𝑓)‘((deg‘𝑓) − 0)))
46	15	nn0cnd 12451	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℂ)
47	46	subid1d 11468	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((deg‘𝑓) − 0) = (deg‘𝑓))
48	47	fveq2d 6832	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((coeff‘𝑓)‘((deg‘𝑓) − 0)) = ((coeff‘𝑓)‘(deg‘𝑓)))
49	45, 48	eqtrd 2768	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0) = ((coeff‘𝑓)‘(deg‘𝑓)))
50	43, 49	eqtrid 2780	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)), 0))‘0) = ((coeff‘𝑓)‘(deg‘𝑓)))
51	28, 32, 50	3eqtrd 2772	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘0) = ((coeff‘𝑓)‘(deg‘𝑓)))
52	12	simprd 495	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ≠ 0𝑝)
53		eqid 2733	. . . . . . . . . . 11 ⊢ (deg‘𝑓) = (deg‘𝑓)
54	53, 18	dgreq0 26199	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
55	13, 54	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
56	55	necon3bid 2973	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
57	52, 56	mpbid 232	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
58	51, 57	eqnetrd 2996	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘0) ≠ 0)
59		ne0p 26140	. . . . . 6 ⊢ ((0 ∈ ℂ ∧ ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘0) ≠ 0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ≠ 0𝑝)
60	25, 58, 59	sylancr 587	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ≠ 0𝑝)
61		eldifsn 4737	. . . . 5 ⊢ ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ∈ (Poly‘ℤ) ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ≠ 0𝑝))
62	24, 60, 61	sylanbrc 583	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ∈ ((Poly‘ℤ) ∖ {0𝑝}))
63		oveq1 7359	. . . . . . . . 9 ⊢ (𝑧 = (1 / 𝐴) → (𝑧↑𝑘) = ((1 / 𝐴)↑𝑘))
64	63	oveq2d 7368	. . . . . . . 8 ⊢ (𝑧 = (1 / 𝐴) → (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
65	64	sumeq2sdv 15612	. . . . . . 7 ⊢ (𝑧 = (1 / 𝐴) → Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
66		eqid 2733	. . . . . . 7 ⊢ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))
67		sumex 15597	. . . . . . 7 ⊢ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)) ∈ V
68	65, 66, 67	fvmpt 6935	. . . . . 6 ⊢ ((1 / 𝐴) ∈ ℂ → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘(1 / 𝐴)) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
69	7, 68	syl 17	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘(1 / 𝐴)) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
70	18	coef3 26165	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Poly‘ℤ) → (coeff‘𝑓):ℕ0⟶ℂ)
71	13, 70	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (coeff‘𝑓):ℕ0⟶ℂ)
72		elfznn0 13522	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...(deg‘𝑓)) → 𝑛 ∈ ℕ0)
73		ffvelcdm 7020	. . . . . . . . . 10 ⊢ (((coeff‘𝑓):ℕ0⟶ℂ ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
74	71, 72, 73	syl2an 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
75	4	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ ℂ)
76		expcl 13988	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐴↑𝑛) ∈ ℂ)
77	75, 72, 76	syl2an 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑛) ∈ ℂ)
78	74, 77	mulcld 11139	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) ∈ ℂ)
79	75, 15	expcld 14055	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
80	79	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
81		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ≠ 0)
82	15	nn0zd 12500	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℤ)
83	75, 81, 82	expne0d 14061	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ≠ 0)
84	83	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ≠ 0)
85	78, 80, 84	divcld 11904	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → ((((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) ∈ ℂ)
86		fveq2 6828	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → ((coeff‘𝑓)‘𝑛) = ((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)))
87		oveq2 7360	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → (𝐴↑𝑛) = (𝐴↑((0 + (deg‘𝑓)) − 𝑘)))
88	86, 87	oveq12d 7370	. . . . . . . 8 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → (((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) = (((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))))
89	88	oveq1d 7367	. . . . . . 7 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → ((((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) = ((((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))) / (𝐴↑(deg‘𝑓))))
90	85, 89	fsumrev2 15691	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → Σ𝑛 ∈ (0...(deg‘𝑓))((((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) = Σ𝑘 ∈ (0...(deg‘𝑓))((((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))) / (𝐴↑(deg‘𝑓))))
91	46	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (deg‘𝑓) ∈ ℂ)
92	91	addlidd 11321	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (0 + (deg‘𝑓)) = (deg‘𝑓))
93	92	oveq1d 7367	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((0 + (deg‘𝑓)) − 𝑘) = ((deg‘𝑓) − 𝑘))
94	93	fveq2d 6832	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) = ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)))
95	93	oveq2d 7368	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑((0 + (deg‘𝑓)) − 𝑘)) = (𝐴↑((deg‘𝑓) − 𝑘)))
96	75	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ∈ ℂ)
97	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ≠ 0)
98		elfznn0 13522	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → 𝑘 ∈ ℕ0)
99	98	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℕ0)
100	99	nn0zd 12500	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℤ)
101	82	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (deg‘𝑓) ∈ ℤ)
102	96, 97, 100, 101	expsubd 14066	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑((deg‘𝑓) − 𝑘)) = ((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)))
103	95, 102	eqtrd 2768	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑((0 + (deg‘𝑓)) − 𝑘)) = ((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)))
104	94, 103	oveq12d 7370	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘))))
105	104	oveq1d 7367	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))) / (𝐴↑(deg‘𝑓))) = ((((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘))) / (𝐴↑(deg‘𝑓))))
106	79	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
107		expcl 13988	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
108	75, 98, 107	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ∈ ℂ)
109	96, 97, 100	expne0d 14061	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ≠ 0)
110	106, 108, 109	divcld 11904	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)) ∈ ℂ)
111	83	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ≠ 0)
112	29, 110, 106, 111	divassd 11939	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘))) / (𝐴↑(deg‘𝑓))) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)) / (𝐴↑(deg‘𝑓)))))
113	106, 111	dividd 11902	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((𝐴↑(deg‘𝑓)) / (𝐴↑(deg‘𝑓))) = 1)
114	113	oveq1d 7367	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (((𝐴↑(deg‘𝑓)) / (𝐴↑(deg‘𝑓))) / (𝐴↑𝑘)) = (1 / (𝐴↑𝑘)))
115	106, 108, 106, 109, 111	divdiv32d 11929	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)) / (𝐴↑(deg‘𝑓))) = (((𝐴↑(deg‘𝑓)) / (𝐴↑(deg‘𝑓))) / (𝐴↑𝑘)))
116	96, 97, 100	exprecd 14063	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((1 / 𝐴)↑𝑘) = (1 / (𝐴↑𝑘)))
117	114, 115, 116	3eqtr4d 2778	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)) / (𝐴↑(deg‘𝑓))) = ((1 / 𝐴)↑𝑘))
118	117	oveq2d 7368	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (((𝐴↑(deg‘𝑓)) / (𝐴↑𝑘)) / (𝐴↑(deg‘𝑓)))) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
119	105, 112, 118	3eqtrd 2772	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))) / (𝐴↑(deg‘𝑓))) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
120	119	sumeq2dv 15611	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → Σ𝑘 ∈ (0...(deg‘𝑓))((((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)) · (𝐴↑((0 + (deg‘𝑓)) − 𝑘))) / (𝐴↑(deg‘𝑓))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
121	90, 120	eqtrd 2768	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → Σ𝑛 ∈ (0...(deg‘𝑓))((((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
122	18, 53	coeid2 26172	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝑓‘𝐴) = Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)))
123	13, 75, 122	syl2anc 584	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)))
124		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
125	123, 124	eqtr3d 2770	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) = 0)
126	125	oveq1d 7367	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) = (0 / (𝐴↑(deg‘𝑓))))
127		fzfid 13882	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0...(deg‘𝑓)) ∈ Fin)
128	127, 79, 78, 83	fsumdivc 15695	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) = Σ𝑛 ∈ (0...(deg‘𝑓))((((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))))
129	79, 83	div0d 11903	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0 / (𝐴↑(deg‘𝑓))) = 0)
130	126, 128, 129	3eqtr3d 2776	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → Σ𝑛 ∈ (0...(deg‘𝑓))((((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)) / (𝐴↑(deg‘𝑓))) = 0)
131	69, 121, 130	3eqtr2d 2774	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘(1 / 𝐴)) = 0)
132		fveq1 6827	. . . . . 6 ⊢ (𝑔 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) → (𝑔‘(1 / 𝐴)) = ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘(1 / 𝐴)))
133	132	eqeq1d 2735	. . . . 5 ⊢ (𝑔 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) → ((𝑔‘(1 / 𝐴)) = 0 ↔ ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘(1 / 𝐴)) = 0))
134	133	rspcev 3573	. . . 4 ⊢ (((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘))) ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)))‘(1 / 𝐴)) = 0) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0)
135	62, 131, 134	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0)
136		elaa 26252	. . 3 ⊢ ((1 / 𝐴) ∈ 𝔸 ↔ ((1 / 𝐴) ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0))
137	7, 135, 136	sylanbrc 583	. 2 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ 𝔸)
138	3, 137	rexlimddv 3140	1 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∃wrex 3057   ∖ cdif 3895   ⊆ wss 3898  ifcif 4474  {csn 4575   class class class wbr 5093   ↦ cmpt 5174  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  ℂcc 11011  0cc0 11013  1c1 11014   + caddc 11016   · cmul 11018   ≤ cle 11154   − cmin 11351   / cdiv 11781  ℕ₀cn0 12388  ℤcz 12475  ...cfz 13409  ↑cexp 13970  Σcsu 15595  0_𝑝c0p 25598  Polycply 26117  coeffccoe 26119  degcdgr 26120  𝔸caa 26250

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-inf 9334  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-rp 12893  df-fz 13410  df-fzo 13557  df-fl 13698  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-rlim 15398  df-sum 15596  df-0p 25599  df-ply 26121  df-coe 26123  df-dgr 26124  df-aa 26251

This theorem is referenced by: (None)