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Theorem cxpeq 25597

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 1194	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝑁 ∈ ℕ)
2		nnm1nn0 12096	. . . . . . . 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 12441	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
5	3, 4	eleqtrdi 2841	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝑁 − 1) ∈ (ℤ_≥‘0))
6		eluzfz1 13084	. . . . . 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 11909	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
9		2re 11869	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
10		simp2 1139	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ∈ ℕ)
11		nndivre 11836	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (2 / 𝑁) ∈ ℝ)
12	9, 10, 11	sylancr 590	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (2 / 𝑁) ∈ ℝ)
13	12	recnd 10826	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (2 / 𝑁) ∈ ℂ)
14		cxpcl 25516	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ (2 / 𝑁) ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
15	8, 13, 14	sylancr 590	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
16	15	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
17		0nn0 12070	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
18		expcl 13618	. . . . . . . 8 ⊢ (((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ 0 ∈ ℕ₀) → ((-1↑_𝑐(2 / 𝑁))↑0) ∈ ℂ)
19	16, 17, 18	sylancl 589	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → ((-1↑_𝑐(2 / 𝑁))↑0) ∈ ℂ)
20	19	mul02d 10995	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (0 · ((-1↑_𝑐(2 / 𝑁))↑0)) = 0)
21		simprl 771	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐴 = 0)
22	21	oveq1d 7206	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐴↑𝑁) = (0↑𝑁))
23		simprr 773	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐴↑𝑁) = 𝐵)
24	1	0expd 13674	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (0↑𝑁) = 0)
25	22, 23, 24	3eqtr3d 2779	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐵 = 0)
26	25	oveq1d 7206	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐵↑_𝑐(1 / 𝑁)) = (0↑_𝑐(1 / 𝑁)))
27		nncn 11803	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
28		nnne0 11829	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
29		reccl 11462	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (1 / 𝑁) ∈ ℂ)
30		recne0 11468	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (1 / 𝑁) ≠ 0)
31	29, 30	0cxpd 25552	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (0↑_𝑐(1 / 𝑁)) = 0)
32	27, 28, 31	syl2anc 587	. . . . . . . . 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 2771	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐵↑_𝑐(1 / 𝑁)) = 0)
35	34	oveq1d 7206	. . . . . 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 2782	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)))
37		oveq2 7199	. . . . . . 7 ⊢ (𝑛 = 0 → ((-1↑_𝑐(2 / 𝑁))↑𝑛) = ((-1↑_𝑐(2 / 𝑁))↑0))
38	37	oveq2d 7207	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)))
39	38	rspceeqv 3542	. . . . 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 587	. . . 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 460	. . 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 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ)
43		simpr 488	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
44		simpl2 1194	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ)
45	44	nnzd 12246	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℤ)
46		explog 25436	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
47	42, 43, 45, 46	syl3anc 1373	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
48	47	eqcomd 2742	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (exp‘(𝑁 · (log‘𝐴))) = (𝐴↑𝑁))
49	10	nncnd 11811	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ∈ ℂ)
50	49	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℂ)
51	42, 43	logcld 25413	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
52	50, 51	mulcld 10818	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝑁 · (log‘𝐴)) ∈ ℂ)
53	44	nnnn0d 12115	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ₀)
54	42, 53	expcld 13681	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) ∈ ℂ)
55	42, 43, 45	expne0d 13687	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) ≠ 0)
56		eflogeq 25444	. . . . . . 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 1373	. . . . . 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 235	. . . . 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 25413	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (log‘(𝐴↑𝑁)) ∈ ℂ)
60	59	adantr 484	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (log‘(𝐴↑𝑁)) ∈ ℂ)
61		ax-icn 10753	. . . . . . . . . . 11 ⊢ i ∈ ℂ
62		2cn 11870	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
63		picn 25303	. . . . . . . . . . . 12 ⊢ π ∈ ℂ
64	62, 63	mulcli 10805	. . . . . . . . . . 11 ⊢ (2 · π) ∈ ℂ
65	61, 64	mulcli 10805	. . . . . . . . . 10 ⊢ (i · (2 · π)) ∈ ℂ
66		zcn 12146	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
67	66	adantl 485	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℂ)
68		mulcl 10778	. . . . . . . . . 10 ⊢ (((i · (2 · π)) ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((i · (2 · π)) · 𝑚) ∈ ℂ)
69	65, 67, 68	sylancr 590	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((i · (2 · π)) · 𝑚) ∈ ℂ)
70	60, 69	addcld 10817	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((log‘(𝐴↑𝑁)) + ((i · (2 · π)) · 𝑚)) ∈ ℂ)
71	50	adantr 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℂ)
72	51	adantr 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (log‘𝐴) ∈ ℂ)
73	10	nnne0d 11845	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ≠ 0)
74	73	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ≠ 0)
75	70, 71, 72, 74	divmuld 11595	. . . . . . 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 6695	. . . . . . . 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 11566	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (1 / 𝑁) ∈ ℂ)
78	77, 60	mulcld 10818	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((1 / 𝑁) · (log‘(𝐴↑𝑁))) ∈ ℂ)
79	13	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (2 / 𝑁) ∈ ℂ)
80	79, 67	mulcld 10818	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((2 / 𝑁) · 𝑚) ∈ ℂ)
81	61, 63	mulcli 10805	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
82		mulcl 10778	. . . . . . . . . . . . 13 ⊢ ((((2 / 𝑁) · 𝑚) ∈ ℂ ∧ (i · π) ∈ ℂ) → (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ)
83	80, 81, 82	sylancl 589	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ)
84		efadd 15618	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 11611	. . . . . . . . . . . . 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 11577	. . . . . . . . . . . . . 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 11610	. . . . . . . . . . . . . . 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 10992	. . . . . . . . . . . . . . . . . 18 ⊢ (i · (2 · π)) = (2 · (i · π))
91	90	oveq1i 7201	. . . . . . . . . . . . . . . . 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 11610	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((2 · (i · π)) / 𝑁) = ((2 / 𝑁) · (i · π)))
95	91, 94	syl5eq 2783	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((i · (2 · π)) / 𝑁) = ((2 / 𝑁) · (i · π)))
96	95	oveq1d 7206	. . . . . . . . . . . . . . 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 11007	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 7209	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 6699	. . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
103	55	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝐴↑𝑁) ≠ 0)
104	102, 103, 77	cxpefd 25554	. . . . . . . . . . . 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 11911	. . . . . . . . . . . . . . 15 ⊢ -1 ≠ 0
107	106	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → -1 ≠ 0)
108		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
109	105, 107, 79, 108	cxpmul2zd 25558	. . . . . . . . . . . . 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 25554	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐((2 / 𝑁) · 𝑚)) = (exp‘(((2 / 𝑁) · 𝑚) · (log‘-1))))
111		logm1 25431	. . . . . . . . . . . . . . . 16 ⊢ (log‘-1) = (i · π)
112	111	oveq2i 7202	. . . . . . . . . . . . . . 15 ⊢ (((2 / 𝑁) · 𝑚) · (log‘-1)) = (((2 / 𝑁) · 𝑚) · (i · π))
113	112	fveq2i 6698	. . . . . . . . . . . . . 14 ⊢ (exp‘(((2 / 𝑁) · 𝑚) · (log‘-1))) = (exp‘(((2 / 𝑁) · 𝑚) · (i · π)))
114	110, 113	eqtrdi 2787	. . . . . . . . . . . . 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 25550	. . . . . . . . . . . . . . 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 25555	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ≠ 0)
119	118	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐(2 / 𝑁)) ≠ 0)
120	115, 119, 108	expclzd 13686	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑𝑚) ∈ ℂ)
121	44	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℕ)
122	108, 121	zmodcld 13430	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℕ₀)
123	115, 122	expcld 13681	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)) ∈ ℂ)
124	122	nn0zd 12245	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℤ)
125	115, 119, 124	expne0d 13687	. . . . . . . . . . . . . 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 13692	. . . . . . . . . . . . . . 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 12246	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℤ)
128		zre 12145	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
129	121	nnrpd 12591	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℝ⁺)
130		moddifz 13421	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → ((𝑚 − (𝑚 mod 𝑁)) / 𝑁) ∈ ℤ)
131	128, 129, 130	syl2an2 686	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((𝑚 − (𝑚 mod 𝑁)) / 𝑁) ∈ ℤ)
132		expmulz 13646	. . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . 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 12117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℂ)
135	67, 134	subcld 11154	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 − (𝑚 mod 𝑁)) ∈ ℂ)
136	135, 71, 74	divcan2d 11575	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑁 · ((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = (𝑚 − (𝑚 mod 𝑁)))
137	136	oveq2d 7207	. . . . . . . . . . . . . . . 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 25594	. . . . . . . . . . . . . . . . . . 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 7206	. . . . . . . . . . . . . . . . 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 13629	. . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 − (𝑚 mod 𝑁))) = 1)
145	126, 144	eqtr3d 2773	. . . . . . . . . . . . . 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 11581	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 7209	. . . . . . . . . . 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 2781	. . . . . . . . . 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 25419	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
151	42, 43, 150	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
152	151	adantr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (exp‘(log‘𝐴)) = 𝐴)
153	149, 152	eqeq12d 2752	. . . . . . . . 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 13431	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)))
155	108, 121, 154	syl2anc 587	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)))
156		eqcom 2743	. . . . . . . . . . . . 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 7199	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑚 mod 𝑁) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) = ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)))
158	157	oveq2d 7207	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 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	syl5bb 286	. . . . . . . . . . . 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 3527	. . . . . . . . . . 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 416	. . . . . . . . . 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 243	. . . . . . . 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 263	. . . . . 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 3193	. . . . 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 7198	. . . . . . 7 ⊢ ((𝐴↑𝑁) = 𝐵 → ((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) = (𝐵↑_𝑐(1 / 𝑁)))
170	169	oveq1d 7206	. . . . . 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 2747	. . . . 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 3206	. . . 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 248	. . 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 3019	. 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 1140	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
176		nnrecre 11837	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
177	176	3ad2ant2 1136	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (1 / 𝑁) ∈ ℝ)
178	177	recnd 10826	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (1 / 𝑁) ∈ ℂ)
179	175, 178	cxpcld 25550	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (𝐵↑_𝑐(1 / 𝑁)) ∈ ℂ)
180	179	adantr 484	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐵↑_𝑐(1 / 𝑁)) ∈ ℂ)
181		elfznn0 13170	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℕ₀)
182		expcl 13618	. . . . . . 7 ⊢ (((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) ∈ ℂ)
183	15, 181, 182	syl2an 599	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) ∈ ℂ)
184	10	adantr 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ)
185	184	nnnn0d 12115	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
186	180, 183, 185	mulexpd 13696	. . . . 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 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝐵 ∈ ℂ)
188		cxproot 25532	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐵↑_𝑐(1 / 𝑁))↑𝑁) = 𝐵)
189	187, 184, 188	syl2anc 587	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → ((𝐵↑_𝑐(1 / 𝑁))↑𝑁) = 𝐵)
190	181	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℕ₀)
191	190	nn0cnd 12117	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℂ)
192	184	nncnd 11811	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℂ)
193	191, 192	mulcomd 10819	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝑛 · 𝑁) = (𝑁 · 𝑛))
194	193	oveq2d 7207	. . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
196	195, 185, 190	expmuld 13684	. . . . . . . 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 13684	. . . . . . . 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 2779	. . . . . . 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 7206	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑𝑛) = (1↑𝑛))
201		elfzelz 13077	. . . . . . . . 9 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℤ)
202	201	adantl 485	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℤ)
203		1exp 13629	. . . . . . . 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 2775	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁) = 1)
206	189, 205	oveq12d 7209	. . . . 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	mulid1d 10815	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐵 · 1) = 𝐵)
208	186, 206, 207	3eqtrd 2775	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))↑𝑁) = 𝐵)
209		oveq1 7198	. . . . 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 2738	. . . 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 250	. . 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 3193	. 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 215	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∃wrex 3052 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 1c1 10695 ici 10696 + caddc 10697 · cmul 10699 − cmin 11027 -cneg 11028 / cdiv 11454 ℕcn 11795 2c2 11850 ℕ₀cn0 12055 ℤcz 12141 ℤ_≥cuz 12403 ℝ⁺crp 12551 ...cfz 13060 mod cmo 13407 ↑cexp 13600 expce 15586 πcpi 15591 logclog 25397 ↑_𝑐ccxp 25398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-fac 13805 df-bc 13834 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-sin 15594 df-cos 15595 df-pi 15597 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-fbas 20314 df-fg 20315 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-nei 21949 df-lp 21987 df-perf 21988 df-cn 22078 df-cnp 22079 df-haus 22166 df-tx 22413 df-hmeo 22606 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-xms 23172 df-ms 23173 df-tms 23174 df-cncf 23729 df-limc 24717 df-dv 24718 df-log 25399 df-cxp 25400

This theorem is referenced by: 1cubr 25679

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