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Theorem cxpeq 26815

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 1191	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝑁 ∈ ℕ)
2		nnm1nn0 12565	. . . . . . . 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 12918	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
5	3, 4	eleqtrdi 2849	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝑁 − 1) ∈ (ℤ_≥‘0))
6		eluzfz1 13568	. . . . . 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 12378	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
9		2re 12338	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
10		simp2 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝑁 ∈ ℕ)
11		nndivre 12305	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (2 / 𝑁) ∈ ℝ)
12	9, 10, 11	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (2 / 𝑁) ∈ ℝ)
13	12	recnd 11287	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (2 / 𝑁) ∈ ℂ)
14		cxpcl 26731	. . . . . . . . . 10 ⊢ ((-1 ∈ ℂ ∧ (2 / 𝑁) ∈ ℂ) → (-1↑_𝑐(2 / 𝑁)) ∈ ℂ)
15	8, 13, 14	sylancr 587	. . . . . . . . 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 12539	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
18		expcl 14117	. . . . . . . 8 ⊢ (((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ 0 ∈ ℕ₀) → ((-1↑_𝑐(2 / 𝑁))↑0) ∈ ℂ)
19	16, 17, 18	sylancl 586	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → ((-1↑_𝑐(2 / 𝑁))↑0) ∈ ℂ)
20	19	mul02d 11457	. . . . . 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 7446	. . . . . . . . . 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 14176	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (0↑𝑁) = 0)
25	22, 23, 24	3eqtr3d 2783	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐵 = 0)
26	25	oveq1d 7446	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐵↑_𝑐(1 / 𝑁)) = (0↑_𝑐(1 / 𝑁)))
27		nncn 12272	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
28		nnne0 12298	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
29		reccl 11927	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (1 / 𝑁) ∈ ℂ)
30		recne0 11933	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (1 / 𝑁) ≠ 0)
31	29, 30	0cxpd 26767	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) → (0↑_𝑐(1 / 𝑁)) = 0)
32	27, 28, 31	syl2anc 584	. . . . . . . . 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 2775	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → (𝐵↑_𝑐(1 / 𝑁)) = 0)
35	34	oveq1d 7446	. . . . . 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 2786	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ (𝐴 = 0 ∧ (𝐴↑𝑁) = 𝐵)) → 𝐴 = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)))
37		oveq2 7439	. . . . . . 7 ⊢ (𝑛 = 0 → ((-1↑_𝑐(2 / 𝑁))↑𝑛) = ((-1↑_𝑐(2 / 𝑁))↑0))
38	37	oveq2d 7447	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛)) = ((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑0)))
39	38	rspceeqv 3645	. . . . 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 584	. . . 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 1190	. . . . . . . 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 1191	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ)
45	44	nnzd 12638	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℤ)
46		explog 26651	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
47	42, 43, 45, 46	syl3anc 1370	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
48	47	eqcomd 2741	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (exp‘(𝑁 · (log‘𝐴))) = (𝐴↑𝑁))
49	10	nncnd 12280	. . . . . . . . 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 26627	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
52	50, 51	mulcld 11279	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝑁 · (log‘𝐴)) ∈ ℂ)
53	44	nnnn0d 12585	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → 𝑁 ∈ ℕ₀)
54	42, 53	expcld 14183	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) ∈ ℂ)
55	42, 43, 45	expne0d 14189	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) → (𝐴↑𝑁) ≠ 0)
56		eflogeq 26659	. . . . . . 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 1370	. . . . . 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 26627	. . . . . . . . . 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 11212	. . . . . . . . . . 11 ⊢ i ∈ ℂ
62		2cn 12339	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
63		picn 26516	. . . . . . . . . . . 12 ⊢ π ∈ ℂ
64	62, 63	mulcli 11266	. . . . . . . . . . 11 ⊢ (2 · π) ∈ ℂ
65	61, 64	mulcli 11266	. . . . . . . . . 10 ⊢ (i · (2 · π)) ∈ ℂ
66		zcn 12616	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℂ)
67	66	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℂ)
68		mulcl 11237	. . . . . . . . . 10 ⊢ (((i · (2 · π)) ∈ ℂ ∧ 𝑚 ∈ ℂ) → ((i · (2 · π)) · 𝑚) ∈ ℂ)
69	65, 67, 68	sylancr 587	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((i · (2 · π)) · 𝑚) ∈ ℂ)
70	60, 69	addcld 11278	. . . . . . . 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 12314	. . . . . . . . 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 12063	. . . . . . 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 6907	. . . . . . . 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 12034	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (1 / 𝑁) ∈ ℂ)
78	77, 60	mulcld 11279	. . . . . . . . . . . 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 11279	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((2 / 𝑁) · 𝑚) ∈ ℂ)
81	61, 63	mulcli 11266	. . . . . . . . . . . . 13 ⊢ (i · π) ∈ ℂ
82		mulcl 11237	. . . . . . . . . . . . 13 ⊢ ((((2 / 𝑁) · 𝑚) ∈ ℂ ∧ (i · π) ∈ ℂ) → (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ)
83	80, 81, 82	sylancl 586	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (((2 / 𝑁) · 𝑚) · (i · π)) ∈ ℂ)
84		efadd 16127	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 12079	. . . . . . . . . . . . 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 12045	. . . . . . . . . . . . . 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 12078	. . . . . . . . . . . . . . 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 11454	. . . . . . . . . . . . . . . . . 18 ⊢ (i · (2 · π)) = (2 · (i · π))
91	90	oveq1i 7441	. . . . . . . . . . . . . . . . 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 12078	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((2 · (i · π)) / 𝑁) = ((2 / 𝑁) · (i · π)))
95	91, 94	eqtrid 2787	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((i · (2 · π)) / 𝑁) = ((2 / 𝑁) · (i · π)))
96	95	oveq1d 7446	. . . . . . . . . . . . . . 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 11469	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 6911	. . . . . . . . . . 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 26769	. . . . . . . . . . . 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 12380	. . . . . . . . . . . . . . 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 26773	. . . . . . . . . . . . 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 26769	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (-1↑_𝑐((2 / 𝑁) · 𝑚)) = (exp‘(((2 / 𝑁) · 𝑚) · (log‘-1))))
111		logm1 26646	. . . . . . . . . . . . . . . 16 ⊢ (log‘-1) = (i · π)
112	111	oveq2i 7442	. . . . . . . . . . . . . . 15 ⊢ (((2 / 𝑁) · 𝑚) · (log‘-1)) = (((2 / 𝑁) · 𝑚) · (i · π))
113	112	fveq2i 6910	. . . . . . . . . . . . . 14 ⊢ (exp‘(((2 / 𝑁) · 𝑚) · (log‘-1))) = (exp‘(((2 / 𝑁) · 𝑚) · (i · π)))
114	110, 113	eqtrdi 2791	. . . . . . . . . . . . 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 26765	. . . . . . . . . . . . . . 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 26770	. . . . . . . . . . . . . . . 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 14188	. . . . . . . . . . . . . 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 13929	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℕ₀)
123	115, 122	expcld 14183	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)) ∈ ℂ)
124	122	nn0zd 12637	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℤ)
125	115, 119, 124	expne0d 14189	. . . . . . . . . . . . . 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 14194	. . . . . . . . . . . . . . 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 12638	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℤ)
128		zre 12615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ ℝ)
129	121	nnrpd 13073	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → 𝑁 ∈ ℝ⁺)
130		moddifz 13920	. . . . . . . . . . . . . . . . . 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 14146	. . . . . . . . . . . . . . . . 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 12587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ ℂ)
135	67, 134	subcld 11618	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 − (𝑚 mod 𝑁)) ∈ ℂ)
136	135, 71, 74	divcan2d 12043	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑁 · ((𝑚 − (𝑚 mod 𝑁)) / 𝑁)) = (𝑚 − (𝑚 mod 𝑁)))
137	136	oveq2d 7447	. . . . . . . . . . . . . . . 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 26812	. . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . 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 14129	. . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → ((-1↑_𝑐(2 / 𝑁))↑(𝑚 − (𝑚 mod 𝑁))) = 1)
145	126, 144	eqtr3d 2777	. . . . . . . . . . . . . 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 12049	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 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 7449	. . . . . . . . . . 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 2785	. . . . . . . . . 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 26633	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
151	42, 43, 150	syl2anc 584	. . . . . . . . . . 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 2751	. . . . . . . . 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 13930	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)))
155	108, 121, 154	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝐴 ≠ 0) ∧ 𝑚 ∈ ℤ) → (𝑚 mod 𝑁) ∈ (0...(𝑁 − 1)))
156		eqcom 2742	. . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑚 mod 𝑁) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) = ((-1↑_𝑐(2 / 𝑁))↑(𝑚 mod 𝑁)))
158	157	oveq2d 7447	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 3622	. . . . . . . . . . 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 3153	. . . . 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 7438	. . . . . . 7 ⊢ ((𝐴↑𝑁) = 𝐵 → ((𝐴↑𝑁)↑_𝑐(1 / 𝑁)) = (𝐵↑_𝑐(1 / 𝑁)))
170	169	oveq1d 7446	. . . . . 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 2746	. . . . 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 3177	. . . 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 3027	. 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 1137	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
176		nnrecre 12306	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
177	176	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (1 / 𝑁) ∈ ℝ)
178	177	recnd 11287	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) → (1 / 𝑁) ∈ ℂ)
179	175, 178	cxpcld 26765	. . . . . . 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 13657	. . . . . . 7 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℕ₀)
182		expcl 14117	. . . . . . 7 ⊢ (((-1↑_𝑐(2 / 𝑁)) ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ((-1↑_𝑐(2 / 𝑁))↑𝑛) ∈ ℂ)
183	15, 181, 182	syl2an 596	. . . . . 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 12585	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℕ₀)
186	180, 183, 185	mulexpd 14198	. . . . 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 26747	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐵↑_𝑐(1 / 𝑁))↑𝑁) = 𝐵)
189	187, 184, 188	syl2anc 584	. . . . . 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 12587	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑛 ∈ ℂ)
192	184	nncnd 12280	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℂ)
193	191, 192	mulcomd 11280	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝑛 · 𝑁) = (𝑁 · 𝑛))
194	193	oveq2d 7447	. . . . . . . 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 14186	. . . . . . . 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 14186	. . . . . . . 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 2783	. . . . . . 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 7446	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑁)↑𝑛) = (1↑𝑛))
201		elfzelz 13561	. . . . . . . . 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 14129	. . . . . . . 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 2779	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((-1↑_𝑐(2 / 𝑁))↑𝑛)↑𝑁) = 1)
206	189, 205	oveq12d 7449	. . . . 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 11276	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐵 · 1) = 𝐵)
208	186, 206, 207	3eqtrd 2779	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝐵 ∈ ℂ) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (((𝐵↑_𝑐(1 / 𝑁)) · ((-1↑_𝑐(2 / 𝑁))↑𝑛))↑𝑁) = 𝐵)
209		oveq1 7438	. . . . 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 2737	. . . 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 3153	. 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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ℝ⁺crp 13032 ...cfz 13544 mod cmo 13906 ↑cexp 14099 expce 16094 πcpi 16099 logclog 26611 ↑_𝑐ccxp 26612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614

This theorem is referenced by: 1cubr 26900

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