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Theorem cxpeq 26818

Description: Solve an equation involving an 𝑁-th power. The expression -1↑_𝑐(2 / 𝑁) = exp(2πi / 𝑁) is a way to write the primitive 𝑁-th root of unity with the smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)

**Assertion**
Ref	Expression
cxpeq	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑁) = 𝐵 ↔ ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝑛,𝑁

Proof of Theorem cxpeq Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1192	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝑁 ∈ ℕ)
2		nnm1nn0 12594	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝑁 − 1) ∈ ℕ₀)
4		nn0uz 12945	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
5	3, 4	eleqtrdi 2854	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝑁 − 1) ∈ (ℤ_≥‘0))
6		eluzfz1 13591	. . . . . 6 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘0) → 0 ∈ (0...(𝑁 − 1)))
7	5, 6	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 0 ∈ (0...(𝑁 − 1)))
8		neg1cn 12407	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
9		2re 12367	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
10		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ∈ ℕ)
11		nndivre 12334	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (2 / 𝑁) ∈ ℝ)
12	9, 10, 11	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (2 / 𝑁) ∈ ℝ)
13	12	recnd 11318	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (2 / 𝑁) ∈ ℂ)
14		cxpcl 26734	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ (2 / 𝑁) ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
15	8, 13, 14	sylancr 586	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
16	15	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
17		0nn0 12568	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
18		expcl 14130	. . . . . . . 8 ⊢ (((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ 0 ∈ ℕ₀) → ((-1↑_𝑐(2 / 𝑁))↑0) ∈ ℂ)
19	16, 17, 18	sylancl 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → ((-1↑_𝑐(2 / 𝑁))↑0) ∈ ℂ)
20	19	mul02d 11488	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (0 · ((-1↑_𝑐(2 / 𝑁))↑0)) = 0)
21		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐴 = 0)
22	21	oveq1d 7463	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐴↑𝑁) = (0↑𝑁))
23		simprr 772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐴↑𝑁) = 𝐵)
24	1	0expd 14189	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (0↑𝑁) = 0)
25	22, 23, 24	3eqtr3d 2788	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐵 = 0)
26	25	oveq1d 7463	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐵↑_𝑐(1 / 𝑁)) = (0↑_𝑐(1 / 𝑁)))
27		nncn 12301	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
28		nnne0 12327	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
29		reccl 11956	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (1 / 𝑁) ∈ ℂ)
30		recne0 11962	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (1 / 𝑁) ≠ 0)
31	29, 30	0cxpd 26770	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (0↑_𝑐(1 / 𝑁)) = 0)
32	27, 28, 31	syl2anc 583	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (0↑_𝑐(1 / 𝑁)) = 0)
33	1, 32	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (0↑_𝑐(1 / 𝑁)) = 0)
34	26, 33	eqtrd 2780	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐵↑_𝑐(1 / 𝑁)) = 0)
35	34	oveq1d 7463	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)) = (0 · ((-1↑_𝑐(2 / 𝑁))↑0)))
36	20, 35, 21	3eqtr4rd 2791	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)))
37		oveq2 7456	. . . . . . 7 ⊢ (𝑛 = 0 → ((-1↑_𝑐(2 / 𝑁))↑𝑛) = ((-1↑_𝑐(2 / 𝑁))↑0))
38	37	oveq2d 7464	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)))
39	38	rspceeqv 3658	. . . . 5 ⊢ ((0 ∈ (0...(𝑁 − 1)) ∧ 𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0))) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)))
40	7, 36, 39	syl2anc 583	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)))
41	40	expr 456	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 = 0) → ((𝐴↑𝑁) = 𝐵 → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
42		simpl1 1191	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
43		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
44		simpl2 1192	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ)
45	44	nnzd 12666	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℤ)
46		explog 26654	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
47	42, 43, 45, 46	syl3anc 1371	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
48	47	eqcomd 2746	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (exp‘(𝑁 · (log‘𝐴))) = (𝐴↑𝑁))
49	10	nncnd 12309	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ∈ ℂ)
50	49	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℂ)
51	42, 43	logcld 26630	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
52	50, 51	mulcld 11310	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝑁 · (log‘𝐴)) ∈ ℂ)
53	44	nnnn0d 12613	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ₀)
54	42, 53	expcld 14196	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) ∈ ℂ)
55	42, 43, 45	expne0d 14202	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) ≠ 0)
56		eflogeq 26662	. . . . . . 7 ⊢ (((𝑁 · (log‘𝐴)) ∈ ℂ ∧ (𝐴↑𝑁) ∈ ℂ ∧ (𝐴↑𝑁) ≠ 0) → ((exp‘(𝑁 · (log‘𝐴))) = (𝐴↑𝑁) ↔ ∃𝑚 ∈ ℤ (𝑁 · (log‘𝐴)) = ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚))))
57	52, 54, 55, 56	syl3anc 1371	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → ((exp‘(𝑁 · (log‘𝐴))) = (𝐴↑𝑁) ↔ ∃𝑚 ∈ ℤ (𝑁 · (log‘𝐴)) = ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚))))
58	48, 57	mpbid 232	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → ∃𝑚 ∈ ℤ (𝑁 · (log‘𝐴)) = ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)))
59	54, 55	logcld 26630	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (log‘(𝐴↑𝑁)) ∈ ℂ)
60	59	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (log‘(𝐴↑𝑁)) ∈ ℂ)
61		ax-icn 11243	. . . . . . . . . . 11 ⊢ i ∈ ℂ
62		2cn 12368	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
63		picn 26519	. . . . . . . . . . . 12 ⊢ π ∈ ℂ
64	62, 63	mulcli 11297	. . . . . . . . . . 11 ⊢ (2 · π) ∈ ℂ
65	61, 64	mulcli 11297	. . . . . . . . . 10 ⊢ (i · (2 · π)) ∈ ℂ
66		zcn 12644	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
67	66	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℂ)
68		mulcl 11268	. . . . . . . . . 10 ⊢ (((i · (2 · π)) ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((i · (2 · π)) · 𝑚) ∈ ℂ)
69	65, 67, 68	sylancr 586	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((i · (2 · π)) · 𝑚) ∈ ℂ)
70	60, 69	addcld 11309	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) ∈ ℂ)
71	50	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℂ)
72	51	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (log‘𝐴) ∈ ℂ)
73	10	nnne0d 12343	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ≠ 0)
74	73	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ≠ 0)
75	70, 71, 72, 74	divmuld 12092	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁) = (log‘𝐴) ↔ (𝑁 · (log‘𝐴)) = ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚))))
76		fveq2 6920	. . . . . . . 8 ⊢ ((((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁) = (log‘𝐴) → (exp‘(((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁)) = (exp‘(log‘𝐴)))
77	71, 74	reccld 12063	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (1 / 𝑁) ∈ ℂ)
78	77, 60	mulcld 11310	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((1 / 𝑁) · (log‘(𝐴↑𝑁))) ∈ ℂ)
79	13	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (2 / 𝑁) ∈ ℂ)
80	79, 67	mulcld 11310	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((2 / 𝑁) · 𝑚) ∈ ℂ)
81	61, 63	mulcli 11297	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
82		mulcl 11268	. . . . . . . . . . . . 13 ⊢ ((((2 / 𝑁) · 𝑚) ∈ ℂ ∧ (i · π) ∈ ℂ) → (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ)
83	80, 81, 82	sylancl 585	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ)
84		efadd 16142	. . . . . . . . . . . 12 ⊢ ((((1 / 𝑁) · (log‘(𝐴↑𝑁))) ∈ ℂ ∧ (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ) → (exp‘(((1 / 𝑁) · (log‘(𝐴↑𝑁))) + (((2 / 𝑁) · 𝑚) · (i · π)))) = ((exp‘((1 / 𝑁) · (log‘(𝐴↑𝑁)))) · (exp‘(((2 / 𝑁) · 𝑚) · (i · π)))))
85	78, 83, 84	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (exp‘(((1 / 𝑁) · (log‘(𝐴↑𝑁))) + (((2 / 𝑁) · 𝑚) · (i · π)))) = ((exp‘((1 / 𝑁) · (log‘(𝐴↑𝑁)))) · (exp‘(((2 / 𝑁) · 𝑚) · (i · π)))))
86	60, 69, 71, 74	divdird 12108	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁) = (((log‘(𝐴↑𝑁)) / 𝑁) + (((i · (2 · π)) · 𝑚) / 𝑁)))
87	60, 71, 74	divrec2d 12074	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((log‘(𝐴↑𝑁)) / 𝑁) = ((1 / 𝑁) · (log‘(𝐴↑𝑁))))
88	65	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (i · (2 · π)) ∈ ℂ)
89	88, 67, 71, 74	div23d 12107	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((i · (2 · π)) · 𝑚) / 𝑁) = (((i · (2 · π)) / 𝑁) · 𝑚))
90	61, 62, 63	mul12i 11485	. . . . . . . . . . . . . . . . . 18 ⊢ (i · (2 · π)) = (2 · (i · π))
91	90	oveq1i 7458	. . . . . . . . . . . . . . . . 17 ⊢ ((i · (2 · π)) / 𝑁) = ((2 · (i · π)) / 𝑁)
92	62	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 2 ∈ ℂ)
93	81	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (i · π) ∈ ℂ)
94	92, 93, 71, 74	div23d 12107	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((2 · (i · π)) / 𝑁) = ((2 / 𝑁) · (i · π)))
95	91, 94	eqtrid 2792	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((i · (2 · π)) / 𝑁) = ((2 / 𝑁) · (i · π)))
96	95	oveq1d 7463	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((i · (2 · π)) / 𝑁) · 𝑚) = (((2 / 𝑁) · (i · π)) · 𝑚))
97	79, 93, 67	mul32d 11500	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((2 / 𝑁) · (i · π)) · 𝑚) = (((2 / 𝑁) · 𝑚) · (i · π)))
98	89, 96, 97	3eqtrd 2784	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((i · (2 · π)) · 𝑚) / 𝑁) = (((2 / 𝑁) · 𝑚) · (i · π)))
99	87, 98	oveq12d 7466	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((log‘(𝐴↑𝑁)) / 𝑁) + (((i · (2 · π)) · 𝑚) / 𝑁)) = (((1 / 𝑁) · (log‘(𝐴↑𝑁))) + (((2 / 𝑁) · 𝑚) · (i · π))))
100	86, 99	eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁) = (((1 / 𝑁) · (log‘(𝐴↑𝑁))) + (((2 / 𝑁) · 𝑚) · (i · π))))
101	100	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (exp‘(((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁)) = (exp‘(((1 / 𝑁) · (log‘(𝐴↑𝑁))) + (((2 / 𝑁) · 𝑚) · (i · π)))))
102	54	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
103	55	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝐴↑𝑁) ≠ 0)
104	102, 103, 77	cxpefd 26772	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) = (exp‘((1 / 𝑁) · (log‘(𝐴↑𝑁)))))
105	8	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → -1 ∈ ℂ)
106		neg1ne0 12409	. . . . . . . . . . . . . . 15 ⊢ -1 ≠ 0
107	106	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → -1 ≠ 0)
108		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
109	105, 107, 79, 108	cxpmul2zd 26776	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐((2 / 𝑁) · 𝑚)) = ((-1↑_𝑐(2 / 𝑁))↑𝑚))
110	105, 107, 80	cxpefd 26772	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐((2 / 𝑁) · 𝑚)) = (exp‘(((2 / 𝑁) · 𝑚) · (log‘-1))))
111		logm1 26649	. . . . . . . . . . . . . . . 16 ⊢ (log‘-1) = (i · π)
112	111	oveq2i 7459	. . . . . . . . . . . . . . 15 ⊢ (((2 / 𝑁) · 𝑚) · (log‘-1)) = (((2 / 𝑁) · 𝑚) · (i · π))
113	112	fveq2i 6923	. . . . . . . . . . . . . 14 ⊢ (exp‘(((2 / 𝑁) · 𝑚) · (log‘-1))) = (exp‘(((2 / 𝑁) · 𝑚) · (i · π)))
114	110, 113	eqtrdi 2796	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐((2 / 𝑁) · 𝑚)) = (exp‘(((2 / 𝑁) · 𝑚) · (i · π))))
115	105, 79	cxpcld 26768	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
116	8	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → -1 ∈ ℂ)
117	106	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → -1 ≠ 0)
118	116, 117, 13	cxpne0d 26773	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ≠ 0)
119	118	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐(2 / 𝑁)) ≠ 0)
120	115, 119, 108	expclzd 14201	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑𝑚) ∈ ℂ)
121	44	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℕ)
122	108, 121	zmodcld 13943	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℕ₀)
123	115, 122	expcld 14196	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)) ∈ ℂ)
124	122	nn0zd 12665	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℤ)
125	115, 119, 124	expne0d 14202	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)) ≠ 0)
126	115, 119, 124, 108	expsubd 14207	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 − (𝑚 mod 𝑁))) = (((-1↑_𝑐(2 / 𝑁))↑𝑚) / ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))))
127	121	nnzd 12666	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℤ)
128		zre 12643	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
129	121	nnrpd 13097	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℝ⁺)
130		moddifz 13934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → ((𝑚 − (𝑚 mod 𝑁)) / 𝑁) ∈ ℤ)
131	128, 129, 130	syl2an2 685	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((𝑚 − (𝑚 mod 𝑁)) / 𝑁) ∈ ℤ)
132		expmulz 14159	. . . . . . . . . . . . . . . . 17 ⊢ ((((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ (-1↑_𝑐(2 / 𝑁)) ≠ 0) ∧ (𝑁 ∈ ℤ ∧ ((𝑚 − (𝑚 mod 𝑁)) / 𝑁) ∈ ℤ)) → ((-1↑_𝑐(2 / 𝑁))↑(𝑁 · ((𝑚 − (𝑚 mod 𝑁)) / 𝑁))) = (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)))
133	115, 119, 127, 131, 132	syl22anc 838	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑁 · ((𝑚 − (𝑚 mod 𝑁)) / 𝑁))) = (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)))
134	122	nn0cnd 12615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℂ)
135	67, 134	subcld 11647	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 − (𝑚 mod 𝑁)) ∈ ℂ)
136	135, 71, 74	divcan2d 12072	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑁 · ((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = (𝑚 − (𝑚 mod 𝑁)))
137	136	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑁 · ((𝑚 − (𝑚 mod 𝑁)) / 𝑁))) = ((-1↑_𝑐(2 / 𝑁))↑(𝑚 − (𝑚 mod 𝑁))))
138		root1id 26815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → ((-1↑_𝑐(2 / 𝑁))↑𝑁) = 1)
139	121, 138	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑𝑁) = 1)
140	139	oveq1d 7463	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = (1↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)))
141		1exp 14142	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 − (𝑚 mod 𝑁)) / 𝑁) ∈ ℤ → (1↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = 1)
142	131, 141	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (1↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = 1)
143	140, 142	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = 1)
144	133, 137, 143	3eqtr3d 2788	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 − (𝑚 mod 𝑁))) = 1)
145	126, 144	eqtr3d 2782	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((-1↑_𝑐(2 / 𝑁))↑𝑚) / ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 1)
146	120, 123, 125, 145	diveq1d 12078	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑𝑚) = ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)))
147	109, 114, 146	3eqtr3rd 2789	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)) = (exp‘(((2 / 𝑁) · 𝑚) · (i · π))))
148	104, 147	oveq12d 7466	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = ((exp‘((1 / 𝑁) · (log‘(𝐴↑𝑁)))) · (exp‘(((2 / 𝑁) · 𝑚) · (i · π)))))
149	85, 101, 148	3eqtr4d 2790	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (exp‘(((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁)) = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))))
150		eflog 26636	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
151	42, 43, 150	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
152	151	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (exp‘(log‘𝐴)) = 𝐴)
153	149, 152	eqeq12d 2756	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((exp‘(((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁)) = (exp‘(log‘𝐴)) ↔ (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 𝐴))
154		zmodfz 13944	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)))
155	108, 121, 154	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)))
156		eqcom 2747	. . . . . . . . . . . . 13 ⊢ (𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) ↔ (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = 𝐴)
157		oveq2 7456	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑚 mod 𝑁) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) = ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)))
158	157	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 mod 𝑁) → (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))))
159	158	eqeq1d 2742	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 mod 𝑁) → ((((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = 𝐴 ↔ (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 𝐴))
160	156, 159	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 mod 𝑁) → (𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) ↔ (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 𝐴))
161	160	rspcev 3635	. . . . . . . . . . 11 ⊢ (((𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)) ∧ (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 𝐴) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)))
162	161	ex 412	. . . . . . . . . 10 ⊢ ((𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)) → ((((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 𝐴 → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
163	155, 162	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁))) = 𝐴 → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
164	153, 163	sylbid 240	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((exp‘(((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁)) = (exp‘(log‘𝐴)) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
165	76, 164	syl5 34	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) / 𝑁) = (log‘𝐴) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
166	75, 165	sylbird 260	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((𝑁 · (log‘𝐴)) = ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
167	166	rexlimdva 3161	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (∃𝑚 ∈ ℤ (𝑁 · (log‘𝐴)) = ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
168	58, 167	mpd 15	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)))
169		oveq1 7455	. . . . . . 7 ⊢ ((𝐴↑𝑁) = 𝐵 → ((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) = (𝐵↑_𝑐(1 / 𝑁)))
170	169	oveq1d 7463	. . . . . 6 ⊢ ((𝐴↑𝑁) = 𝐵 → (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)))
171	170	eqeq2d 2751	. . . . 5 ⊢ ((𝐴↑𝑁) = 𝐵 → (𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) ↔ 𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
172	171	rexbidv 3185	. . . 4 ⊢ ((𝐴↑𝑁) = 𝐵 → (∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = (((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) ↔ ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
173	168, 172	syl5ibcom 245	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → ((𝐴↑𝑁) = 𝐵 → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
174	41, 173	pm2.61dane 3035	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑁) = 𝐵 → ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))
175		simp3 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
176		nnrecre 12335	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
177	176	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (1 / 𝑁) ∈ ℝ)
178	177	recnd 11318	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (1 / 𝑁) ∈ ℂ)
179	175, 178	cxpcld 26768	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (𝐵↑_𝑐(1 / 𝑁)) ∈ ℂ)
180	179	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐵↑_𝑐(1 / 𝑁)) ∈ ℂ)
181		elfznn0 13677	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℕ₀)
182		expcl 14130	. . . . . . 7 ⊢ (((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) ∈ ℂ)
183	15, 181, 182	syl2an 595	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) ∈ ℂ)
184	10	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ)
185	184	nnnn0d 12613	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
186	180, 183, 185	mulexpd 14211	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))↑𝑁) = (((𝐵↑_𝑐(1 / 𝑁))↑𝑁) · (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁)))
187	175	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝐵 ∈ ℂ)
188		cxproot 26750	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐵↑_𝑐(1 / 𝑁))↑𝑁) = 𝐵)
189	187, 184, 188	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((𝐵↑_𝑐(1 / 𝑁))↑𝑁) = 𝐵)
190	181	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℕ₀)
191	190	nn0cnd 12615	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℂ)
192	184	nncnd 12309	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℂ)
193	191, 192	mulcomd 11311	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝑛 · 𝑁) = (𝑁 · 𝑛))
194	193	oveq2d 7464	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((-1↑_𝑐(2 / 𝑁))↑(𝑛 · 𝑁)) = ((-1↑_𝑐(2 / 𝑁))↑(𝑁 · 𝑛)))
195	15	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
196	195, 185, 190	expmuld 14199	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((-1↑_𝑐(2 / 𝑁))↑(𝑛 · 𝑁)) = (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁))
197	195, 190, 185	expmuld 14199	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((-1↑_𝑐(2 / 𝑁))↑(𝑁 · 𝑛)) = (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑𝑛))
198	194, 196, 197	3eqtr3d 2788	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁) = (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑𝑛))
199	184, 138	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((-1↑_𝑐(2 / 𝑁))↑𝑁) = 1)
200	199	oveq1d 7463	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑𝑛) = (1↑𝑛))
201		elfzelz 13584	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℤ)
202	201	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℤ)
203		1exp 14142	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
204	202, 203	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (1↑𝑛) = 1)
205	198, 200, 204	3eqtrd 2784	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁) = 1)
206	189, 205	oveq12d 7466	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((𝐵↑_𝑐(1 / 𝑁))↑𝑁) · (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁)) = (𝐵 · 1))
207	187	mulridd 11307	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐵 · 1) = 𝐵)
208	186, 206, 207	3eqtrd 2784	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))↑𝑁) = 𝐵)
209		oveq1 7455	. . . . 5 ⊢ (𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) → (𝐴↑𝑁) = (((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))↑𝑁))
210	209	eqeq1d 2742	. . . 4 ⊢ (𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) → ((𝐴↑𝑁) = 𝐵 ↔ (((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))↑𝑁) = 𝐵))
211	208, 210	syl5ibrcom 247	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) → (𝐴↑𝑁) = 𝐵))
212	211	rexlimdva 3161	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) → (𝐴↑𝑁) = 𝐵))
213	174, 212	impbid 212	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑁) = 𝐵 ↔ ∃𝑛 ∈ (0...(𝑁 − 1))𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 ici 11186 + caddc 11187 · cmul 11189 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 ...cfz 13567 mod cmo 13920 ↑cexp 14112 expce 16109 πcpi 16114 logclog 26614 ↑_𝑐ccxp 26615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-cxp 26617

This theorem is referenced by: 1cubr 26903

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