Theorem aareccl 26635

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 26621	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
2	1	simprbi 503	. . 3 ⊢ (𝐴 ∈ 𝔸 → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
3	2	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)
4		aacn 26622	. . . . 5 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
5		reccl 11962	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
6	4, 5	sylan 592	. . . 4 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ)
7	6	adantr 486	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ ℂ)
8		zsscn 12682	. . . . . . 7 ⊢ ℤ ⊆ ℂ
9	8	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ℤ ⊆ ℂ)
10		simprl 783	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}))
11		eldifsn 4748	. . . . . . . . 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 500	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ∈ (Poly‘ℤ))
14		dgrcl 26532	. . . . . . 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 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑓 ∈ (Poly‘ℤ))
17		0z 12685	. . . . . . . 8 ⊢ 0 ∈ ℤ
18		eqid 2761	. . . . . . . . 9 ⊢ (coeff‘𝑓) = (coeff‘𝑓)
19	18	coef2 26530	. . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑓):ℕ0⟶ℤ)
20	16, 17, 19	sylancl 598	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (coeff‘𝑓):ℕ0⟶ℤ)
21		fznn0sub 13670	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
22	21	adantl 487	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((deg‘𝑓) − 𝑘) ∈ ℕ0)
23	20, 22	ffvelcdmd 7077	. . . . . 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 26500	. . . . 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 11279	. . . . . 6 ⊢ 0 ∈ ℂ
26		eqid 2761	. . . . . . . . . 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 26547	. . . . . . . . 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 12785	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) ∈ ℂ)
30		eqidd 2762	. . . . . . . . . 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 26541	. . . . . . . . 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 6879	. . . . . . . 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 12602	. . . . . . . . . 10 ⊢ 0 ∈ ℕ0
34		breq1 5106	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 ≤ (deg‘𝑓) ↔ 0 ≤ (deg‘𝑓)))
35		oveq2 7420	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((deg‘𝑓) − 𝑘) = ((deg‘𝑓) − 0))
36	35	fveq2d 6881	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) = ((coeff‘𝑓)‘((deg‘𝑓) − 0)))
37	34, 36	ifbieq1d 4507	. . . . . . . . . . 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 2761	. . . . . . . . . . 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 6890	. . . . . . . . . . . 12 ⊢ ((coeff‘𝑓)‘((deg‘𝑓) − 0)) ∈ V
40		c0ex 11281	. . . . . . . . . . . 12 ⊢ 0 ∈ V
41	39, 40	ifex 4533	. . . . . . . . . . 11 ⊢ if(0 ≤ (deg‘𝑓), ((coeff‘𝑓)‘((deg‘𝑓) − 0)), 0) ∈ V
42	37, 38, 41	fvmpt 6985	. . . . . . . . . 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 12651	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 0 ≤ (deg‘𝑓))
45	44	iftrued 4490	. . . . . . . . . 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 12650	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℂ)
47	46	subid1d 11639	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ((deg‘𝑓) − 0) = (deg‘𝑓))
48	47	fveq2d 6881	. . . . . . . . . 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 2796	. . . . . . . . 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 2808	. . . . . . . 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 2800	. . . . . . 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 501	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝑓 ≠ 0𝑝)
53		eqid 2761	. . . . . . . . . . 11 ⊢ (deg‘𝑓) = (deg‘𝑓)
54	53, 18	dgreq0 26564	. . . . . . . . . 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 3000	. . . . . . . 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 3023	. . . . . 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 26505	. . . . . 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 599	. . . . 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 4748	. . . . 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 595	. . . 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 7419	. . . . . . . . 9 ⊢ (𝑧 = (1 / 𝐴) → (𝑧↑𝑘) = ((1 / 𝐴)↑𝑘))
64	63	oveq2d 7428	. . . . . . . 8 ⊢ (𝑧 = (1 / 𝐴) → (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · (𝑧↑𝑘)) = (((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)))
65	64	sumeq2sdv 15850	. . . . . . 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 2761	. . . . . . 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 15835	. . . . . . 7 ⊢ Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘((deg‘𝑓) − 𝑘)) · ((1 / 𝐴)↑𝑘)) ∈ V
68	65, 66, 67	fvmpt 6985	. . . . . 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 26531	. . . . . . . . . . 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 13734	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...(deg‘𝑓)) → 𝑛 ∈ ℕ0)
73		ffvelcdm 7073	. . . . . . . . . 10 ⊢ (((coeff‘𝑓):ℕ0⟶ℂ ∧ 𝑛 ∈ ℕ0) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
74	71, 72, 73	syl2an 608	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → ((coeff‘𝑓)‘𝑛) ∈ ℂ)
75	4	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ∈ ℂ)
76		expcl 14202	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐴↑𝑛) ∈ ℂ)
77	75, 72, 76	syl2an 608	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑛) ∈ ℂ)
78	74, 77	mulcld 11310	. . . . . . . 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 14269	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
80	79	adantr 486	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑛 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
81		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → 𝐴 ≠ 0)
82	15	nn0zd 12699	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (deg‘𝑓) ∈ ℤ)
83	75, 81, 82	expne0d 14275	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝐴↑(deg‘𝑓)) ≠ 0)
84	83	adantr 486	. . . . . . . 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 12074	. . . . . . 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 6877	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → ((coeff‘𝑓)‘𝑛) = ((coeff‘𝑓)‘((0 + (deg‘𝑓)) − 𝑘)))
87		oveq2 7420	. . . . . . . . 9 ⊢ (𝑛 = ((0 + (deg‘𝑓)) − 𝑘) → (𝐴↑𝑛) = (𝐴↑((0 + (deg‘𝑓)) − 𝑘)))
88	86, 87	oveq12d 7430	. . . . . . . 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 7427	. . . . . . 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 15928	. . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (deg‘𝑓) ∈ ℂ)
92	91	addlidd 11492	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (0 + (deg‘𝑓)) = (deg‘𝑓))
93	92	oveq1d 7427	. . . . . . . . . . 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 6881	. . . . . . . . . 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 7428	. . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ∈ ℂ)
97	81	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝐴 ≠ 0)
98		elfznn0 13734	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(deg‘𝑓)) → 𝑘 ∈ ℕ0)
99	98	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℕ0)
100	99	nn0zd 12699	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → 𝑘 ∈ ℤ)
101	82	adantr 486	. . . . . . . . . . . 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 14280	. . . . . . . . . . 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 2796	. . . . . . . . . 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 7430	. . . . . . . . 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 7427	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑(deg‘𝑓)) ∈ ℂ)
107		expcl 14202	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ)
108	75, 98, 107	syl2an 608	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ∈ ℂ)
109	96, 97, 100	expne0d 14275	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) ∧ 𝑘 ∈ (0...(deg‘𝑓))) → (𝐴↑𝑘) ≠ 0)
110	106, 108, 109	divcld 12074	. . . . . . . . 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 486	. . . . . . . . 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 12109	. . . . . . . 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 12072	. . . . . . . . . . 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 7427	. . . . . . . . . 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 12099	. . . . . . . . . 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 14277	. . . . . . . . . 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 2806	. . . . . . . . 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 7428	. . . . . . . 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 2800	. . . . . . 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 15849	. . . . . 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 2796	. . . . 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 26538	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ ℂ) → (𝑓‘𝐴) = Σ𝑛 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑛) · (𝐴↑𝑛)))
123	13, 75, 122	syl2anc 596	. . . . . . . 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 785	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (𝑓‘𝐴) = 0)
125	123, 124	eqtr3d 2798	. . . . . . 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 7427	. . . . . 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 14096	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0...(deg‘𝑓)) ∈ Fin)
128	127, 79, 78, 83	fsumdivc 15932	. . . . . 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 12073	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (0 / (𝐴↑(deg‘𝑓))) = 0)
130	126, 128, 129	3eqtr3d 2804	. . . . 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 2802	. . . 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 6876	. . . . . 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 2763	. . . . 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 3577	. . . 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 596	. . 3 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0)
136		elaa 26621	. . 3 ⊢ ((1 / 𝐴) ∈ 𝔸 ↔ ((1 / 𝐴) ∈ ℂ ∧ ∃𝑔 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑔‘(1 / 𝐴)) = 0))
137	7, 135, 136	sylanbrc 595	. 2 ⊢ (((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) ∧ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) ∧ (𝑓‘𝐴) = 0)) → (1 / 𝐴) ∈ 𝔸)
138	3, 137	rexlimddv 3170	1 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896   ⊆ wss 3899  ifcif 4482  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  ℂcc 11179  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186   ≤ cle 11325   − cmin 11522   / cdiv 11954  ℕ₀cn0 12587  ℤcz 12674  ...cfz 13620  ↑cexp 14184  Σcsu 15833  0_𝑝c0p 25970  Polycply 26482  coeffccoe 26484  degcdgr 26485  𝔸caa 26619

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-rlim 15636  df-sum 15834  df-0p 25971  df-ply 26486  df-coe 26488  df-dgr 26489  df-aa 26620

This theorem is used by: (None)