Step |
Hyp |
Ref |
Expression |
1 |
|
1cubr.r |
|- R = { 1 , ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) } |
2 |
|
ax-1cn |
|- 1 e. CC |
3 |
|
neg1cn |
|- -u 1 e. CC |
4 |
|
ax-icn |
|- _i e. CC |
5 |
|
3cn |
|- 3 e. CC |
6 |
|
sqrtcl |
|- ( 3 e. CC -> ( sqrt ` 3 ) e. CC ) |
7 |
5 6
|
ax-mp |
|- ( sqrt ` 3 ) e. CC |
8 |
4 7
|
mulcli |
|- ( _i x. ( sqrt ` 3 ) ) e. CC |
9 |
3 8
|
addcli |
|- ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) e. CC |
10 |
|
halfcl |
|- ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) e. CC -> ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC ) |
11 |
9 10
|
ax-mp |
|- ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC |
12 |
3 8
|
subcli |
|- ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) e. CC |
13 |
|
halfcl |
|- ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) e. CC -> ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC ) |
14 |
12 13
|
ax-mp |
|- ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC |
15 |
2 11 14
|
3pm3.2i |
|- ( 1 e. CC /\ ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC /\ ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC ) |
16 |
2
|
elexi |
|- 1 e. _V |
17 |
|
ovex |
|- ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. _V |
18 |
|
ovex |
|- ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. _V |
19 |
16 17 18
|
tpss |
|- ( ( 1 e. CC /\ ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC /\ ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) e. CC ) <-> { 1 , ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) } C_ CC ) |
20 |
15 19
|
mpbi |
|- { 1 , ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) } C_ CC |
21 |
1 20
|
eqsstri |
|- R C_ CC |
22 |
21
|
sseli |
|- ( A e. R -> A e. CC ) |
23 |
22
|
pm4.71ri |
|- ( A e. R <-> ( A e. CC /\ A e. R ) ) |
24 |
|
3nn |
|- 3 e. NN |
25 |
|
cxpeq |
|- ( ( A e. CC /\ 3 e. NN /\ 1 e. CC ) -> ( ( A ^ 3 ) = 1 <-> E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) ) |
26 |
24 2 25
|
mp3an23 |
|- ( A e. CC -> ( ( A ^ 3 ) = 1 <-> E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) ) |
27 |
|
eltpg |
|- ( A e. CC -> ( A e. { 1 , ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) } <-> ( A = 1 \/ A = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) \/ A = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) ) ) |
28 |
1
|
eleq2i |
|- ( A e. R <-> A e. { 1 , ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) } ) |
29 |
|
3m1e2 |
|- ( 3 - 1 ) = 2 |
30 |
|
2cn |
|- 2 e. CC |
31 |
30
|
addid2i |
|- ( 0 + 2 ) = 2 |
32 |
29 31
|
eqtr4i |
|- ( 3 - 1 ) = ( 0 + 2 ) |
33 |
32
|
oveq2i |
|- ( 0 ... ( 3 - 1 ) ) = ( 0 ... ( 0 + 2 ) ) |
34 |
|
0z |
|- 0 e. ZZ |
35 |
|
fztp |
|- ( 0 e. ZZ -> ( 0 ... ( 0 + 2 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) } ) |
36 |
34 35
|
ax-mp |
|- ( 0 ... ( 0 + 2 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) } |
37 |
33 36
|
eqtri |
|- ( 0 ... ( 3 - 1 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) } |
38 |
37
|
rexeqi |
|- ( E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) |
39 |
|
3ne0 |
|- 3 =/= 0 |
40 |
5 39
|
reccli |
|- ( 1 / 3 ) e. CC |
41 |
|
1cxp |
|- ( ( 1 / 3 ) e. CC -> ( 1 ^c ( 1 / 3 ) ) = 1 ) |
42 |
40 41
|
ax-mp |
|- ( 1 ^c ( 1 / 3 ) ) = 1 |
43 |
42
|
oveq1i |
|- ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) |
44 |
43
|
eqeq2i |
|- ( A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) |
45 |
44
|
rexbii |
|- ( E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) |
46 |
34
|
elexi |
|- 0 e. _V |
47 |
|
ovex |
|- ( 0 + 1 ) e. _V |
48 |
|
ovex |
|- ( 0 + 2 ) e. _V |
49 |
|
oveq2 |
|- ( n = 0 -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 0 ) ) |
50 |
30 5 39
|
divcli |
|- ( 2 / 3 ) e. CC |
51 |
|
cxpcl |
|- ( ( -u 1 e. CC /\ ( 2 / 3 ) e. CC ) -> ( -u 1 ^c ( 2 / 3 ) ) e. CC ) |
52 |
3 50 51
|
mp2an |
|- ( -u 1 ^c ( 2 / 3 ) ) e. CC |
53 |
|
exp0 |
|- ( ( -u 1 ^c ( 2 / 3 ) ) e. CC -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ 0 ) = 1 ) |
54 |
52 53
|
ax-mp |
|- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 0 ) = 1 |
55 |
49 54
|
eqtrdi |
|- ( n = 0 -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = 1 ) |
56 |
55
|
oveq2d |
|- ( n = 0 -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. 1 ) ) |
57 |
|
1t1e1 |
|- ( 1 x. 1 ) = 1 |
58 |
56 57
|
eqtrdi |
|- ( n = 0 -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = 1 ) |
59 |
58
|
eqeq2d |
|- ( n = 0 -> ( A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = 1 ) ) |
60 |
|
id |
|- ( n = ( 0 + 1 ) -> n = ( 0 + 1 ) ) |
61 |
2
|
addid2i |
|- ( 0 + 1 ) = 1 |
62 |
60 61
|
eqtrdi |
|- ( n = ( 0 + 1 ) -> n = 1 ) |
63 |
62
|
oveq2d |
|- ( n = ( 0 + 1 ) -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 1 ) ) |
64 |
|
exp1 |
|- ( ( -u 1 ^c ( 2 / 3 ) ) e. CC -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ 1 ) = ( -u 1 ^c ( 2 / 3 ) ) ) |
65 |
52 64
|
ax-mp |
|- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 1 ) = ( -u 1 ^c ( 2 / 3 ) ) |
66 |
63 65
|
eqtrdi |
|- ( n = ( 0 + 1 ) -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( -u 1 ^c ( 2 / 3 ) ) ) |
67 |
66
|
oveq2d |
|- ( n = ( 0 + 1 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. ( -u 1 ^c ( 2 / 3 ) ) ) ) |
68 |
52
|
mulid2i |
|- ( 1 x. ( -u 1 ^c ( 2 / 3 ) ) ) = ( -u 1 ^c ( 2 / 3 ) ) |
69 |
|
1cubrlem |
|- ( ( -u 1 ^c ( 2 / 3 ) ) = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) /\ ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) |
70 |
69
|
simpli |
|- ( -u 1 ^c ( 2 / 3 ) ) = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) |
71 |
68 70
|
eqtri |
|- ( 1 x. ( -u 1 ^c ( 2 / 3 ) ) ) = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) |
72 |
67 71
|
eqtrdi |
|- ( n = ( 0 + 1 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) |
73 |
72
|
eqeq2d |
|- ( n = ( 0 + 1 ) -> ( A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) ) |
74 |
|
id |
|- ( n = ( 0 + 2 ) -> n = ( 0 + 2 ) ) |
75 |
74 31
|
eqtrdi |
|- ( n = ( 0 + 2 ) -> n = 2 ) |
76 |
75
|
oveq2d |
|- ( n = ( 0 + 2 ) -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) |
77 |
76
|
oveq2d |
|- ( n = ( 0 + 2 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) ) |
78 |
52
|
sqcli |
|- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) e. CC |
79 |
78
|
mulid2i |
|- ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) |
80 |
69
|
simpri |
|- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) |
81 |
79 80
|
eqtri |
|- ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) |
82 |
77 81
|
eqtrdi |
|- ( n = ( 0 + 2 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) |
83 |
82
|
eqeq2d |
|- ( n = ( 0 + 2 ) -> ( A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) ) |
84 |
46 47 48 59 73 83
|
rextp |
|- ( E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> ( A = 1 \/ A = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) \/ A = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) ) |
85 |
38 45 84
|
3bitri |
|- ( E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> ( A = 1 \/ A = ( ( -u 1 + ( _i x. ( sqrt ` 3 ) ) ) / 2 ) \/ A = ( ( -u 1 - ( _i x. ( sqrt ` 3 ) ) ) / 2 ) ) ) |
86 |
27 28 85
|
3bitr4g |
|- ( A e. CC -> ( A e. R <-> E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) ) |
87 |
26 86
|
bitr4d |
|- ( A e. CC -> ( ( A ^ 3 ) = 1 <-> A e. R ) ) |
88 |
87
|
pm5.32i |
|- ( ( A e. CC /\ ( A ^ 3 ) = 1 ) <-> ( A e. CC /\ A e. R ) ) |
89 |
23 88
|
bitr4i |
|- ( A e. R <-> ( A e. CC /\ ( A ^ 3 ) = 1 ) ) |