Theorem aareccl 26448

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 26438	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
2	1	simprbi 502	. . 3 ⊢ (𝐴 ∈ 𝔸 → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
3	2	adantr 485	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
4		aacn 26439	. . . . 5 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
5		reccl 11867	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
6	4, 5	sylan 591	. . . 4 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
7	6	adantr 485	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ ℂ)
8		zsscn 12590	. . . . . . 7 ⊢ ℤ ⊆ ℂ
9	8	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ℤ ⊆ ℂ)
10		simprl 782	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
11		eldifsn 4749	. . . . . . . . 9 ⊢ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ↔ (𝑓 ∈ (Poly‘ℤ) ∧ 𝑓 ≠ 0𝑝))
12	10, 11	sylib 221	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 ∈ (Poly‘ℤ) ∧ 𝑓 ≠ 0𝑝))
13	12	simpld 499	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ (Poly‘ℤ))
14		dgrcl 26351	. . . . . . 7 ⊢ (𝑓 ∈ (Poly‘ℤ) → (deg‘𝑓) ∈ ℕ0)
15	13, 14	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℕ0)
16	13	adantr 485	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑓 ∈ (Poly‘ℤ))
17		0z 12593	. . . . . . . 8 ⊢ 0 ∈ ℤ
18		eqid 2765	. . . . . . . . 9 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
19	18	coef2 26349	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑓):ℕ0⟶ℤ)
20	16, 17, 19	sylancl 597	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (coeff‘𝑓):ℕ0⟶ℤ)
21		fznn0sub 13575	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
22	21	adantl 486	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
23	20, 22	ffvelcdmd 7070	. . . . . 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 26320	. . . . 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 11186	. . . . . 6 ⊢ 0 ∈ ℂ
26		eqid 2765	. . . . . . . . . 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 26366	. . . . . . . . 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 18	. . . . . . . 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 12692	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) ∈ ℂ)
30		eqidd 2766	. . . . . . . . . 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 26360	. . . . . . . . 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 6873	. . . . . . . 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 12510	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
34		breq1 5108	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
35		oveq2 7408	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((deg‘𝑓) − 𝑘) = ((deg‘𝑓) − 0))
36	35	fveq2d 6875	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) = ((coeff‘𝑓)‘((deg‘𝑓) − 0)))
37	34, 36	ifbieq1d 4508	. . . . . . . . . . 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 2765	. . . . . . . . . . 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 6884	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑓)‘((deg‘𝑓) − 0)) ∈ V
40		c0ex 11188	. . . . . . . . . . . 12 ⊢ 0 ∈ V
41	39, 40	ifex 4534	. . . . . . . . . . 11 ⊢ if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0) ∈ V
42	37, 38, 41	fvmpt 6979	. . . . . . . . . 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 12559	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 0 ≤ (deg‘𝑓))
45	44	iftrued 4491	. . . . . . . . . 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 12558	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℂ)
47	46	subid1d 11546	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((deg‘𝑓) − 0) = (deg‘𝑓))
48	47	fveq2d 6875	. . . . . . . . . 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 2800	. . . . . . . . 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 2812	. . . . . . . 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 2804	. . . . . . 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 500	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ≠ 0𝑝)
53		eqid 2765	. . . . . . . . . . 11 ⊢ (deg‘𝑓) = (deg‘𝑓)
54	53, 18	dgreq0 26383	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Poly‘ℤ) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
55	13, 54	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
56	55	necon3bid 3004	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
57	52, 56	mpbid 235	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
58	51, 57	eqnetrd 3027	. . . . . 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 26325	. . . . . 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 598	. . . . 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 4749	. . . . 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 594	. . . 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 7407	. . . . . . . . 9 ⊢ (𝑧 = (1 / 𝐴) → (𝑧↑𝑘) = ((1 / 𝐴)↑𝑘))
64	63	oveq2d 7416	. . . . . . . 8 ⊢ (𝑧 = (1 / 𝐴) → (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
65	64	sumeq2sdv 15744	. . . . . . 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 2765	. . . . . . 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 15729	. . . . . . 7 ⊢ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)) ∈ V
68	65, 66, 67	fvmpt 6979	. . . . . 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 18	. . . . 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 26350	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (Poly‘ℤ) → (coeff‘𝑓):ℕ0⟶ℂ)
71	13, 70	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (coeff‘𝑓):ℕ0⟶ℂ)
72		elfznn0 13639	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...(deg‘𝑓)) → 𝑛 ∈ ℕ0)
73		ffvelcdm 7066	. . . . . . . . . 10 ⊢ (((coeff‘𝑓):ℕ0⟶ℂ ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
74	71, 72, 73	syl2an 607	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
75	4	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ ℂ)
76		expcl 14106	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐴↑𝑛) ∈ ℂ)
77	75, 72, 76	syl2an 607	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑛) ∈ ℂ)
78	74, 77	mulcld 11217	. . . . . . . 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 14173	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
80	79	adantr 485	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
81		simplr 780	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ≠ 0)
82	15	nn0zd 12607	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℤ)
83	75, 81, 82	expne0d 14179	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ≠ 0)
84	83	adantr 485	. . . . . . . 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 11982	. . . . . . 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 6871	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → ((coeff‘𝑓)‘𝑛) = ((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)))
87		oveq2 7408	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → (𝐴↑𝑛) = (𝐴↑((0 + (deg‘𝑓)) − 𝑘)))
88	86, 87	oveq12d 7418	. . . . . . . 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 7415	. . . . . . 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 15823	. . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (deg‘𝑓) ∈ ℂ)
92	91	addlidd 11399	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (0 + (deg‘𝑓)) = (deg‘𝑓))
93	92	oveq1d 7415	. . . . . . . . . . 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 6875	. . . . . . . . . 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 7416	. . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ∈ ℂ)
97	81	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ≠ 0)
98		elfznn0 13639	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → 𝑘 ∈ ℕ0)
99	98	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℕ0)
100	99	nn0zd 12607	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℤ)
101	82	adantr 485	. . . . . . . . . . . 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 14184	. . . . . . . . . . 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 2800	. . . . . . . . . 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 7418	. . . . . . . . 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 7415	. . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
107		expcl 14106	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
108	75, 98, 107	syl2an 607	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ∈ ℂ)
109	96, 97, 100	expne0d 14179	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ≠ 0)
110	106, 108, 109	divcld 11982	. . . . . . . . 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 485	. . . . . . . . 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 12017	. . . . . . . 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 11980	. . . . . . . . . . 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 7415	. . . . . . . . . 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 12007	. . . . . . . . . 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 14181	. . . . . . . . . 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 2810	. . . . . . . . 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 7416	. . . . . . . 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 2804	. . . . . . 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 15743	. . . . . 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 2800	. . . . 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 26357	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝑓‘𝐴) = Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)))
123	13, 75, 122	syl2anc 595	. . . . . . . 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 784	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
125	123, 124	eqtr3d 2802	. . . . . . 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 7415	. . . . . 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 14000	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0...(deg‘𝑓)) ∈ Fin)
128	127, 79, 78, 83	fsumdivc 15827	. . . . . 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 11981	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0 / (𝐴↑(deg‘𝑓))) = 0)
130	126, 128, 129	3eqtr3d 2808	. . . . 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 2806	. . . 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 6870	. . . . . 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 2767	. . . . 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 3584	. . . 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 595	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0)
136		elaa 26438	. . 3 ⊢ ((1 / 𝐴) ∈ 𝔸 ↔ ((1 / 𝐴) ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0))
137	7, 135, 136	sylanbrc 594	. 2 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ 𝔸)
138	3, 137	rexlimddv 3172	1 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089   ∖ cdif 3904   ⊆ wss 3907  ifcif 4483  {csn 4585   class class class wbr 5105   ↦ cmpt 5186  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  0cc0 11088  1c1 11089   + caddc 11091   · cmul 11093   ≤ cle 11232   − cmin 11429   / cdiv 11859  ℕ₀cn0 12495  ℤcz 12582  ...cfz 13526  ↑cexp 14088  Σcsu 15727  0_𝑝c0p 25789  Polycply 26302  coeffccoe 26304  degcdgr 26305  𝔸caa 26436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-rp 13008  df-fz 13527  df-fzo 13674  df-fl 13816  df-seq 14029  df-exp 14089  df-hash 14358  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-rlim 15530  df-sum 15728  df-0p 25790  df-ply 26306  df-coe 26308  df-dgr 26309  df-aa 26437

This theorem is referenced by: (None)