Step |
Hyp |
Ref |
Expression |
1 |
|
proot1ex.g |
⊢ 𝐺 = ( ( mulGrp ‘ ℂfld ) ↾s ( ℂ ∖ { 0 } ) ) |
2 |
|
proot1ex.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
3 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
4 |
|
2rp |
⊢ 2 ∈ ℝ+ |
5 |
|
nnrp |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ+ ) |
6 |
|
rpdivcl |
⊢ ( ( 2 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+ ) → ( 2 / 𝑁 ) ∈ ℝ+ ) |
7 |
4 5 6
|
sylancr |
⊢ ( 𝑁 ∈ ℕ → ( 2 / 𝑁 ) ∈ ℝ+ ) |
8 |
7
|
rpcnd |
⊢ ( 𝑁 ∈ ℕ → ( 2 / 𝑁 ) ∈ ℂ ) |
9 |
|
cxpcl |
⊢ ( ( - 1 ∈ ℂ ∧ ( 2 / 𝑁 ) ∈ ℂ ) → ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ℂ ) |
10 |
3 8 9
|
sylancr |
⊢ ( 𝑁 ∈ ℕ → ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ℂ ) |
11 |
3
|
a1i |
⊢ ( 𝑁 ∈ ℕ → - 1 ∈ ℂ ) |
12 |
|
neg1ne0 |
⊢ - 1 ≠ 0 |
13 |
12
|
a1i |
⊢ ( 𝑁 ∈ ℕ → - 1 ≠ 0 ) |
14 |
11 13 8
|
cxpne0d |
⊢ ( 𝑁 ∈ ℕ → ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ≠ 0 ) |
15 |
|
eldifsn |
⊢ ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ↔ ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ℂ ∧ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ≠ 0 ) ) |
16 |
10 14 15
|
sylanbrc |
⊢ ( 𝑁 ∈ ℕ → ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ) |
17 |
3
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → - 1 ∈ ℂ ) |
18 |
12
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → - 1 ≠ 0 ) |
19 |
|
nn0cn |
⊢ ( 𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ ) |
20 |
|
mulcl |
⊢ ( ( ( 2 / 𝑁 ) ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( ( 2 / 𝑁 ) · 𝑥 ) ∈ ℂ ) |
21 |
8 19 20
|
syl2an |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( 2 / 𝑁 ) · 𝑥 ) ∈ ℂ ) |
22 |
17 18 21
|
cxpefd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( - 1 ↑𝑐 ( ( 2 / 𝑁 ) · 𝑥 ) ) = ( exp ‘ ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ) ) |
23 |
22
|
eqeq1d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( - 1 ↑𝑐 ( ( 2 / 𝑁 ) · 𝑥 ) ) = 1 ↔ ( exp ‘ ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ) = 1 ) ) |
24 |
|
logcl |
⊢ ( ( - 1 ∈ ℂ ∧ - 1 ≠ 0 ) → ( log ‘ - 1 ) ∈ ℂ ) |
25 |
3 12 24
|
mp2an |
⊢ ( log ‘ - 1 ) ∈ ℂ |
26 |
|
mulcl |
⊢ ( ( ( ( 2 / 𝑁 ) · 𝑥 ) ∈ ℂ ∧ ( log ‘ - 1 ) ∈ ℂ ) → ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ∈ ℂ ) |
27 |
21 25 26
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ∈ ℂ ) |
28 |
|
efeq1 |
⊢ ( ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ∈ ℂ → ( ( exp ‘ ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ) = 1 ↔ ( ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) / ( i · ( 2 · π ) ) ) ∈ ℤ ) ) |
29 |
27 28
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( exp ‘ ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) ) = 1 ↔ ( ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) / ( i · ( 2 · π ) ) ) ∈ ℤ ) ) |
30 |
|
2cn |
⊢ 2 ∈ ℂ |
31 |
30
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 2 ∈ ℂ ) |
32 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
33 |
32
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 𝑁 ∈ ℂ ) |
34 |
19
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 𝑥 ∈ ℂ ) |
35 |
|
nnne0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 ) |
36 |
35
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 𝑁 ≠ 0 ) |
37 |
31 33 34 36
|
div13d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( 2 / 𝑁 ) · 𝑥 ) = ( ( 𝑥 / 𝑁 ) · 2 ) ) |
38 |
|
logm1 |
⊢ ( log ‘ - 1 ) = ( i · π ) |
39 |
38
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( log ‘ - 1 ) = ( i · π ) ) |
40 |
37 39
|
oveq12d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) = ( ( ( 𝑥 / 𝑁 ) · 2 ) · ( i · π ) ) ) |
41 |
34 33 36
|
divcld |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 / 𝑁 ) ∈ ℂ ) |
42 |
|
ax-icn |
⊢ i ∈ ℂ |
43 |
|
picn |
⊢ π ∈ ℂ |
44 |
42 43
|
mulcli |
⊢ ( i · π ) ∈ ℂ |
45 |
44
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( i · π ) ∈ ℂ ) |
46 |
41 31 45
|
mulassd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( 𝑥 / 𝑁 ) · 2 ) · ( i · π ) ) = ( ( 𝑥 / 𝑁 ) · ( 2 · ( i · π ) ) ) ) |
47 |
42
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → i ∈ ℂ ) |
48 |
43
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → π ∈ ℂ ) |
49 |
31 47 48
|
mul12d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 2 · ( i · π ) ) = ( i · ( 2 · π ) ) ) |
50 |
49
|
oveq2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( 𝑥 / 𝑁 ) · ( 2 · ( i · π ) ) ) = ( ( 𝑥 / 𝑁 ) · ( i · ( 2 · π ) ) ) ) |
51 |
40 46 50
|
3eqtrd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) = ( ( 𝑥 / 𝑁 ) · ( i · ( 2 · π ) ) ) ) |
52 |
51
|
oveq1d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) / ( i · ( 2 · π ) ) ) = ( ( ( 𝑥 / 𝑁 ) · ( i · ( 2 · π ) ) ) / ( i · ( 2 · π ) ) ) ) |
53 |
30 43
|
mulcli |
⊢ ( 2 · π ) ∈ ℂ |
54 |
42 53
|
mulcli |
⊢ ( i · ( 2 · π ) ) ∈ ℂ |
55 |
54
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( i · ( 2 · π ) ) ∈ ℂ ) |
56 |
|
ine0 |
⊢ i ≠ 0 |
57 |
|
2ne0 |
⊢ 2 ≠ 0 |
58 |
|
pire |
⊢ π ∈ ℝ |
59 |
|
pipos |
⊢ 0 < π |
60 |
58 59
|
gt0ne0ii |
⊢ π ≠ 0 |
61 |
30 43 57 60
|
mulne0i |
⊢ ( 2 · π ) ≠ 0 |
62 |
42 53 56 61
|
mulne0i |
⊢ ( i · ( 2 · π ) ) ≠ 0 |
63 |
62
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( i · ( 2 · π ) ) ≠ 0 ) |
64 |
41 55 63
|
divcan4d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( 𝑥 / 𝑁 ) · ( i · ( 2 · π ) ) ) / ( i · ( 2 · π ) ) ) = ( 𝑥 / 𝑁 ) ) |
65 |
52 64
|
eqtrd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) / ( i · ( 2 · π ) ) ) = ( 𝑥 / 𝑁 ) ) |
66 |
65
|
eleq1d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( ( ( ( 2 / 𝑁 ) · 𝑥 ) · ( log ‘ - 1 ) ) / ( i · ( 2 · π ) ) ) ∈ ℤ ↔ ( 𝑥 / 𝑁 ) ∈ ℤ ) ) |
67 |
23 29 66
|
3bitrd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( - 1 ↑𝑐 ( ( 2 / 𝑁 ) · 𝑥 ) ) = 1 ↔ ( 𝑥 / 𝑁 ) ∈ ℤ ) ) |
68 |
8
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 2 / 𝑁 ) ∈ ℂ ) |
69 |
|
simpr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 𝑥 ∈ ℕ0 ) |
70 |
17 68 69
|
cxpmul2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( - 1 ↑𝑐 ( ( 2 / 𝑁 ) · 𝑥 ) ) = ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ↑ 𝑥 ) ) |
71 |
|
cnfldexp |
⊢ ( ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ℂ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 ( .g ‘ ( mulGrp ‘ ℂfld ) ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ↑ 𝑥 ) ) |
72 |
10 71
|
sylan |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 ( .g ‘ ( mulGrp ‘ ℂfld ) ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ↑ 𝑥 ) ) |
73 |
|
cnring |
⊢ ℂfld ∈ Ring |
74 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
75 |
|
cnfld0 |
⊢ 0 = ( 0g ‘ ℂfld ) |
76 |
|
cndrng |
⊢ ℂfld ∈ DivRing |
77 |
74 75 76
|
drngui |
⊢ ( ℂ ∖ { 0 } ) = ( Unit ‘ ℂfld ) |
78 |
|
eqid |
⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld ) |
79 |
77 78
|
unitsubm |
⊢ ( ℂfld ∈ Ring → ( ℂ ∖ { 0 } ) ∈ ( SubMnd ‘ ( mulGrp ‘ ℂfld ) ) ) |
80 |
73 79
|
mp1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ℂ ∖ { 0 } ) ∈ ( SubMnd ‘ ( mulGrp ‘ ℂfld ) ) ) |
81 |
16
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ) |
82 |
|
eqid |
⊢ ( .g ‘ ( mulGrp ‘ ℂfld ) ) = ( .g ‘ ( mulGrp ‘ ℂfld ) ) |
83 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
84 |
82 1 83
|
submmulg |
⊢ ( ( ( ℂ ∖ { 0 } ) ∈ ( SubMnd ‘ ( mulGrp ‘ ℂfld ) ) ∧ 𝑥 ∈ ℕ0 ∧ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 ( .g ‘ ( mulGrp ‘ ℂfld ) ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) ) |
85 |
80 69 81 84
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 ( .g ‘ ( mulGrp ‘ ℂfld ) ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) ) |
86 |
70 72 85
|
3eqtr2rd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = ( - 1 ↑𝑐 ( ( 2 / 𝑁 ) · 𝑥 ) ) ) |
87 |
86
|
eqeq1d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 1 ↔ ( - 1 ↑𝑐 ( ( 2 / 𝑁 ) · 𝑥 ) ) = 1 ) ) |
88 |
|
nnz |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) |
89 |
88
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 𝑁 ∈ ℤ ) |
90 |
|
nn0z |
⊢ ( 𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ ) |
91 |
90
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → 𝑥 ∈ ℤ ) |
92 |
|
dvdsval2 |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝑥 ∈ ℤ ) → ( 𝑁 ∥ 𝑥 ↔ ( 𝑥 / 𝑁 ) ∈ ℤ ) ) |
93 |
89 36 91 92
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑁 ∥ 𝑥 ↔ ( 𝑥 / 𝑁 ) ∈ ℤ ) ) |
94 |
67 87 93
|
3bitr4rd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑥 ∈ ℕ0 ) → ( 𝑁 ∥ 𝑥 ↔ ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 1 ) ) |
95 |
94
|
ralrimiva |
⊢ ( 𝑁 ∈ ℕ → ∀ 𝑥 ∈ ℕ0 ( 𝑁 ∥ 𝑥 ↔ ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 1 ) ) |
96 |
77 1
|
unitgrp |
⊢ ( ℂfld ∈ Ring → 𝐺 ∈ Grp ) |
97 |
73 96
|
mp1i |
⊢ ( 𝑁 ∈ ℕ → 𝐺 ∈ Grp ) |
98 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
99 |
77 1
|
unitgrpbas |
⊢ ( ℂ ∖ { 0 } ) = ( Base ‘ 𝐺 ) |
100 |
|
cnfld1 |
⊢ 1 = ( 1r ‘ ℂfld ) |
101 |
77 1 100
|
unitgrpid |
⊢ ( ℂfld ∈ Ring → 1 = ( 0g ‘ 𝐺 ) ) |
102 |
73 101
|
ax-mp |
⊢ 1 = ( 0g ‘ 𝐺 ) |
103 |
99 2 83 102
|
odeq |
⊢ ( ( 𝐺 ∈ Grp ∧ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 = ( 𝑂 ‘ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) ↔ ∀ 𝑥 ∈ ℕ0 ( 𝑁 ∥ 𝑥 ↔ ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 1 ) ) ) |
104 |
97 16 98 103
|
syl3anc |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 = ( 𝑂 ‘ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) ↔ ∀ 𝑥 ∈ ℕ0 ( 𝑁 ∥ 𝑥 ↔ ( 𝑥 ( .g ‘ 𝐺 ) ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 1 ) ) ) |
105 |
95 104
|
mpbird |
⊢ ( 𝑁 ∈ ℕ → 𝑁 = ( 𝑂 ‘ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) ) |
106 |
105
|
eqcomd |
⊢ ( 𝑁 ∈ ℕ → ( 𝑂 ‘ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 𝑁 ) |
107 |
99 2
|
odf |
⊢ 𝑂 : ( ℂ ∖ { 0 } ) ⟶ ℕ0 |
108 |
|
ffn |
⊢ ( 𝑂 : ( ℂ ∖ { 0 } ) ⟶ ℕ0 → 𝑂 Fn ( ℂ ∖ { 0 } ) ) |
109 |
107 108
|
ax-mp |
⊢ 𝑂 Fn ( ℂ ∖ { 0 } ) |
110 |
|
fniniseg |
⊢ ( 𝑂 Fn ( ℂ ∖ { 0 } ) → ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ∧ ( 𝑂 ‘ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 𝑁 ) ) ) |
111 |
109 110
|
mp1i |
⊢ ( 𝑁 ∈ ℕ → ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ℂ ∖ { 0 } ) ∧ ( 𝑂 ‘ ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ) = 𝑁 ) ) ) |
112 |
16 106 111
|
mpbir2and |
⊢ ( 𝑁 ∈ ℕ → ( - 1 ↑𝑐 ( 2 / 𝑁 ) ) ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) |