Step |
Hyp |
Ref |
Expression |
1 |
|
ral0 |
|- A. i e. (/) ( W ` i ) = ( W ` 0 ) |
2 |
|
oveq2 |
|- ( ( # ` W ) = 0 -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ 0 ) ) |
3 |
|
fzo0 |
|- ( 0 ..^ 0 ) = (/) |
4 |
2 3
|
eqtrdi |
|- ( ( # ` W ) = 0 -> ( 0 ..^ ( # ` W ) ) = (/) ) |
5 |
4
|
raleqdv |
|- ( ( # ` W ) = 0 -> ( A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) <-> A. i e. (/) ( W ` i ) = ( W ` 0 ) ) ) |
6 |
1 5
|
mpbiri |
|- ( ( # ` W ) = 0 -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |
7 |
6
|
a1d |
|- ( ( # ` W ) = 0 -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) ) |
8 |
|
simprl |
|- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> W e. Word V ) |
9 |
|
lencl |
|- ( W e. Word V -> ( # ` W ) e. NN0 ) |
10 |
|
1nn0 |
|- 1 e. NN0 |
11 |
10
|
a1i |
|- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> 1 e. NN0 ) |
12 |
|
df-ne |
|- ( ( # ` W ) =/= 0 <-> -. ( # ` W ) = 0 ) |
13 |
|
elnnne0 |
|- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. NN0 /\ ( # ` W ) =/= 0 ) ) |
14 |
13
|
simplbi2com |
|- ( ( # ` W ) =/= 0 -> ( ( # ` W ) e. NN0 -> ( # ` W ) e. NN ) ) |
15 |
12 14
|
sylbir |
|- ( -. ( # ` W ) = 0 -> ( ( # ` W ) e. NN0 -> ( # ` W ) e. NN ) ) |
16 |
15
|
adantr |
|- ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> ( ( # ` W ) e. NN0 -> ( # ` W ) e. NN ) ) |
17 |
16
|
impcom |
|- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> ( # ` W ) e. NN ) |
18 |
|
neqne |
|- ( -. ( # ` W ) = 1 -> ( # ` W ) =/= 1 ) |
19 |
18
|
ad2antll |
|- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> ( # ` W ) =/= 1 ) |
20 |
|
nngt1ne1 |
|- ( ( # ` W ) e. NN -> ( 1 < ( # ` W ) <-> ( # ` W ) =/= 1 ) ) |
21 |
17 20
|
syl |
|- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> ( 1 < ( # ` W ) <-> ( # ` W ) =/= 1 ) ) |
22 |
19 21
|
mpbird |
|- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> 1 < ( # ` W ) ) |
23 |
|
elfzo0 |
|- ( 1 e. ( 0 ..^ ( # ` W ) ) <-> ( 1 e. NN0 /\ ( # ` W ) e. NN /\ 1 < ( # ` W ) ) ) |
24 |
11 17 22 23
|
syl3anbrc |
|- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> 1 e. ( 0 ..^ ( # ` W ) ) ) |
25 |
24
|
ex |
|- ( ( # ` W ) e. NN0 -> ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> 1 e. ( 0 ..^ ( # ` W ) ) ) ) |
26 |
9 25
|
syl |
|- ( W e. Word V -> ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> 1 e. ( 0 ..^ ( # ` W ) ) ) ) |
27 |
26
|
adantr |
|- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> 1 e. ( 0 ..^ ( # ` W ) ) ) ) |
28 |
27
|
impcom |
|- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> 1 e. ( 0 ..^ ( # ` W ) ) ) |
29 |
|
simprr |
|- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> ( W cyclShift 1 ) = W ) |
30 |
|
lbfzo0 |
|- ( 0 e. ( 0 ..^ ( # ` W ) ) <-> ( # ` W ) e. NN ) |
31 |
30 13
|
sylbbr |
|- ( ( ( # ` W ) e. NN0 /\ ( # ` W ) =/= 0 ) -> 0 e. ( 0 ..^ ( # ` W ) ) ) |
32 |
31
|
ex |
|- ( ( # ` W ) e. NN0 -> ( ( # ` W ) =/= 0 -> 0 e. ( 0 ..^ ( # ` W ) ) ) ) |
33 |
12 32
|
syl5bir |
|- ( ( # ` W ) e. NN0 -> ( -. ( # ` W ) = 0 -> 0 e. ( 0 ..^ ( # ` W ) ) ) ) |
34 |
9 33
|
syl |
|- ( W e. Word V -> ( -. ( # ` W ) = 0 -> 0 e. ( 0 ..^ ( # ` W ) ) ) ) |
35 |
34
|
adantr |
|- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> ( -. ( # ` W ) = 0 -> 0 e. ( 0 ..^ ( # ` W ) ) ) ) |
36 |
35
|
com12 |
|- ( -. ( # ` W ) = 0 -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> 0 e. ( 0 ..^ ( # ` W ) ) ) ) |
37 |
36
|
adantr |
|- ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> 0 e. ( 0 ..^ ( # ` W ) ) ) ) |
38 |
37
|
imp |
|- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> 0 e. ( 0 ..^ ( # ` W ) ) ) |
39 |
|
elfzoelz |
|- ( 1 e. ( 0 ..^ ( # ` W ) ) -> 1 e. ZZ ) |
40 |
|
cshweqrep |
|- ( ( W e. Word V /\ 1 e. ZZ ) -> ( ( ( W cyclShift 1 ) = W /\ 0 e. ( 0 ..^ ( # ` W ) ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) ) |
41 |
39 40
|
sylan2 |
|- ( ( W e. Word V /\ 1 e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( W cyclShift 1 ) = W /\ 0 e. ( 0 ..^ ( # ` W ) ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) ) |
42 |
41
|
imp |
|- ( ( ( W e. Word V /\ 1 e. ( 0 ..^ ( # ` W ) ) ) /\ ( ( W cyclShift 1 ) = W /\ 0 e. ( 0 ..^ ( # ` W ) ) ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) |
43 |
8 28 29 38 42
|
syl22anc |
|- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) |
44 |
|
0nn0 |
|- 0 e. NN0 |
45 |
|
fzossnn0 |
|- ( 0 e. NN0 -> ( 0 ..^ ( # ` W ) ) C_ NN0 ) |
46 |
|
ssralv |
|- ( ( 0 ..^ ( # ` W ) ) C_ NN0 -> ( A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) ) |
47 |
44 45 46
|
mp2b |
|- ( A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) |
48 |
|
eqcom |
|- ( ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) <-> ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` 0 ) ) |
49 |
|
elfzoelz |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> i e. ZZ ) |
50 |
|
zre |
|- ( i e. ZZ -> i e. RR ) |
51 |
|
ax-1rid |
|- ( i e. RR -> ( i x. 1 ) = i ) |
52 |
50 51
|
syl |
|- ( i e. ZZ -> ( i x. 1 ) = i ) |
53 |
52
|
oveq2d |
|- ( i e. ZZ -> ( 0 + ( i x. 1 ) ) = ( 0 + i ) ) |
54 |
|
zcn |
|- ( i e. ZZ -> i e. CC ) |
55 |
54
|
addid2d |
|- ( i e. ZZ -> ( 0 + i ) = i ) |
56 |
53 55
|
eqtrd |
|- ( i e. ZZ -> ( 0 + ( i x. 1 ) ) = i ) |
57 |
49 56
|
syl |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( 0 + ( i x. 1 ) ) = i ) |
58 |
57
|
oveq1d |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) = ( i mod ( # ` W ) ) ) |
59 |
|
zmodidfzoimp |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( i mod ( # ` W ) ) = i ) |
60 |
58 59
|
eqtrd |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) = i ) |
61 |
60
|
fveqeq2d |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` 0 ) <-> ( W ` i ) = ( W ` 0 ) ) ) |
62 |
61
|
biimpd |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` 0 ) -> ( W ` i ) = ( W ` 0 ) ) ) |
63 |
48 62
|
syl5bi |
|- ( i e. ( 0 ..^ ( # ` W ) ) -> ( ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> ( W ` i ) = ( W ` 0 ) ) ) |
64 |
63
|
ralimia |
|- ( A. i e. ( 0 ..^ ( # ` W ) ) ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |
65 |
47 64
|
syl |
|- ( A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |
66 |
43 65
|
syl |
|- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |
67 |
66
|
ex |
|- ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) ) |
68 |
67
|
impancom |
|- ( ( -. ( # ` W ) = 0 /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> ( -. ( # ` W ) = 1 -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) ) |
69 |
|
eqid |
|- ( W ` 0 ) = ( W ` 0 ) |
70 |
|
c0ex |
|- 0 e. _V |
71 |
|
fveqeq2 |
|- ( i = 0 -> ( ( W ` i ) = ( W ` 0 ) <-> ( W ` 0 ) = ( W ` 0 ) ) ) |
72 |
70 71
|
ralsn |
|- ( A. i e. { 0 } ( W ` i ) = ( W ` 0 ) <-> ( W ` 0 ) = ( W ` 0 ) ) |
73 |
69 72
|
mpbir |
|- A. i e. { 0 } ( W ` i ) = ( W ` 0 ) |
74 |
|
oveq2 |
|- ( ( # ` W ) = 1 -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ 1 ) ) |
75 |
|
fzo01 |
|- ( 0 ..^ 1 ) = { 0 } |
76 |
74 75
|
eqtrdi |
|- ( ( # ` W ) = 1 -> ( 0 ..^ ( # ` W ) ) = { 0 } ) |
77 |
76
|
raleqdv |
|- ( ( # ` W ) = 1 -> ( A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) <-> A. i e. { 0 } ( W ` i ) = ( W ` 0 ) ) ) |
78 |
73 77
|
mpbiri |
|- ( ( # ` W ) = 1 -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |
79 |
68 78
|
pm2.61d2 |
|- ( ( -. ( # ` W ) = 0 /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |
80 |
79
|
ex |
|- ( -. ( # ` W ) = 0 -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) ) |
81 |
7 80
|
pm2.61i |
|- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0 ..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) |