Theorem aareccl 26317

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 26307	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
2	1	simprbi 498	. . 3 ⊢ (𝐴 ∈ 𝔸 → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
3	2	adantr 481	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
4		aacn 26308	. . . . 5 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
5		reccl 11814	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
6	4, 5	sylan 586	. . . 4 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
7	6	adantr 481	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ ℂ)
8		zsscn 12530	. . . . . . 7 ⊢ ℤ ⊆ ℂ
9	8	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ℤ ⊆ ℂ)
10		simprl 776	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
11		eldifsn 4726	. . . . . . . . 9 ⊢ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ (𝑓 ∈ (Poly‘ℤ) ∧ 𝑓 ≠ 0𝑝))
12	10, 11	sylib 219	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ (Poly‘ℤ) ∧ 𝑓 ≠ 0𝑝))
13	12	simpld 495	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ (Poly‘ℤ))
14		dgrcl 26223	. . . . . . 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 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑓 ∈ (Poly‘ℤ))
17		0z 12533	. . . . . . . 8 ⊢ 0 ∈ ℤ
18		eqid 2740	. . . . . . . . 9 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
19	18	coef2 26221	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑓):ℕ0⟶ℤ)
20	16, 17, 19	sylancl 592	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (coeff‘𝑓):ℕ0⟶ℤ)
21		fznn0sub 13508	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
22	21	adantl 482	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
23	20, 22	ffvelcdmd 7033	. . . . . 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 26192	. . . . 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 11134	. . . . . 6 ⊢ 0 ∈ ℂ
26		eqid 2740	. . . . . . . . . 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 26238	. . . . . . . . 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 12632	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) ∈ ℂ)
30		eqidd 2741	. . . . . . . . . 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 26232	. . . . . . . . 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 6836	. . . . . . . 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 12450	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
34		breq1 5082	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
35		oveq2 7371	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((deg‘𝑓) − 𝑘) = ((deg‘𝑓) − 0))
36	35	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) = ((coeff‘𝑓)‘((deg‘𝑓) − 0)))
37	34, 36	ifbieq1d 4486	. . . . . . . . . . 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 2740	. . . . . . . . . . 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 6847	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑓)‘((deg‘𝑓) − 0)) ∈ V
40		c0ex 11136	. . . . . . . . . . . 12 ⊢ 0 ∈ V
41	39, 40	ifex 4512	. . . . . . . . . . 11 ⊢ if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0) ∈ V
42	37, 38, 41	fvmpt 6942	. . . . . . . . . 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 12499	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 0 ≤ (deg‘𝑓))
45	44	iftrued 4469	. . . . . . . . . 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 12498	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℂ)
47	46	subid1d 11492	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((deg‘𝑓) − 0) = (deg‘𝑓))
48	47	fveq2d 6838	. . . . . . . . . 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 2775	. . . . . . . . 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 2787	. . . . . . . 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 2779	. . . . . . 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 496	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ≠ 0𝑝)
53		eqid 2740	. . . . . . . . . . 11 ⊢ (deg‘𝑓) = (deg‘𝑓)
54	53, 18	dgreq0 26255	. . . . . . . . . 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 2979	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
57	52, 56	mpbid 233	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
58	51, 57	eqnetrd 3002	. . . . . 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 26197	. . . . . 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 593	. . . . 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 4726	. . . . 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 589	. . . 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 7370	. . . . . . . . 9 ⊢ (𝑧 = (1 / 𝐴) → (𝑧↑𝑘) = ((1 / 𝐴)↑𝑘))
64	63	oveq2d 7379	. . . . . . . 8 ⊢ (𝑧 = (1 / 𝐴) → (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
65	64	sumeq2sdv 15663	. . . . . . 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 2740	. . . . . . 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 15648	. . . . . . 7 ⊢ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)) ∈ V
68	65, 66, 67	fvmpt 6942	. . . . . 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 26222	. . . . . . . . . . 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 13572	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...(deg‘𝑓)) → 𝑛 ∈ ℕ0)
73		ffvelcdm 7029	. . . . . . . . . 10 ⊢ (((coeff‘𝑓):ℕ0⟶ℂ ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
74	71, 72, 73	syl2an 602	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
75	4	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ ℂ)
76		expcl 14039	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐴↑𝑛) ∈ ℂ)
77	75, 72, 76	syl2an 602	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑛) ∈ ℂ)
78	74, 77	mulcld 11163	. . . . . . . 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 14106	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
80	79	adantr 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
81		simplr 774	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ≠ 0)
82	15	nn0zd 12547	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℤ)
83	75, 81, 82	expne0d 14112	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ≠ 0)
84	83	adantr 481	. . . . . . . 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 11929	. . . . . . 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 6834	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → ((coeff‘𝑓)‘𝑛) = ((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)))
87		oveq2 7371	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → (𝐴↑𝑛) = (𝐴↑((0 + (deg‘𝑓)) − 𝑘)))
88	86, 87	oveq12d 7381	. . . . . . . 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 7378	. . . . . . 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 15742	. . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (deg‘𝑓) ∈ ℂ)
92	91	addlidd 11345	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (0 + (deg‘𝑓)) = (deg‘𝑓))
93	92	oveq1d 7378	. . . . . . . . . . 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 6838	. . . . . . . . . 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 7379	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ∈ ℂ)
97	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ≠ 0)
98		elfznn0 13572	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → 𝑘 ∈ ℕ0)
99	98	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℕ0)
100	99	nn0zd 12547	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℤ)
101	82	adantr 481	. . . . . . . . . . . 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 14117	. . . . . . . . . . 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 2775	. . . . . . . . . 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 7381	. . . . . . . . 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 7378	. . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
107		expcl 14039	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
108	75, 98, 107	syl2an 602	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ∈ ℂ)
109	96, 97, 100	expne0d 14112	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ≠ 0)
110	106, 108, 109	divcld 11929	. . . . . . . . 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 481	. . . . . . . . 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 11964	. . . . . . . 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 11927	. . . . . . . . . . 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 7378	. . . . . . . . . 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 11954	. . . . . . . . . 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 14114	. . . . . . . . . 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 2785	. . . . . . . . 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 7379	. . . . . . . 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 2779	. . . . . . 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 15662	. . . . . 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 2775	. . . . 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 26229	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝑓‘𝐴) = Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)))
123	13, 75, 122	syl2anc 590	. . . . . . . 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 778	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
125	123, 124	eqtr3d 2777	. . . . . . 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 7378	. . . . . 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 13933	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0...(deg‘𝑓)) ∈ Fin)
128	127, 79, 78, 83	fsumdivc 15746	. . . . . 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 11928	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0 / (𝐴↑(deg‘𝑓))) = 0)
130	126, 128, 129	3eqtr3d 2783	. . . . 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 2781	. . . 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 6833	. . . . . 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 2742	. . . . 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 3567	. . . 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 590	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0)
136		elaa 26307	. . 3 ⊢ ((1 / 𝐴) ∈ 𝔸 ↔ ((1 / 𝐴) ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0))
137	7, 135, 136	sylanbrc 589	. 2 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ 𝔸)
138	3, 137	rexlimddv 3147	1 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064   ∖ cdif 3887   ⊆ wss 3890  ifcif 4461  {csn 4562   class class class wbr 5079   ↦ cmpt 5160  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  ℂcc 11034  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041   ≤ cle 11178   − cmin 11375   / cdiv 11805  ℕ₀cn0 12435  ℤcz 12522  ...cfz 13459  ↑cexp 14021  Σcsu 15646  0_𝑝c0p 25661  Polycply 26174  coeffccoe 26176  degcdgr 26177  𝔸caa 26305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-map 8772  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-inf 9353  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-fz 13460  df-fzo 13607  df-fl 13749  df-seq 13962  df-exp 14022  df-hash 14291  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-clim 15448  df-rlim 15449  df-sum 15647  df-0p 25662  df-ply 26178  df-coe 26180  df-dgr 26181  df-aa 26306

This theorem is referenced by: (None)