Step |
Hyp |
Ref |
Expression |
1 |
|
elfzo0 |
|- ( I e. ( 0 ..^ ( # ` W ) ) <-> ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) ) |
2 |
|
nn0z |
|- ( I e. NN0 -> I e. ZZ ) |
3 |
2
|
3ad2ant1 |
|- ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) -> I e. ZZ ) |
4 |
|
zsubcl |
|- ( ( I e. ZZ /\ N e. ZZ ) -> ( I - N ) e. ZZ ) |
5 |
3 4
|
sylan |
|- ( ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) /\ N e. ZZ ) -> ( I - N ) e. ZZ ) |
6 |
|
simpl2 |
|- ( ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) /\ N e. ZZ ) -> ( # ` W ) e. NN ) |
7 |
5 6
|
jca |
|- ( ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) /\ N e. ZZ ) -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) |
8 |
7
|
ex |
|- ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) -> ( N e. ZZ -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) ) |
9 |
1 8
|
sylbi |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> ( N e. ZZ -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) ) |
10 |
9
|
impcom |
|- ( ( N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) |
11 |
10
|
3adant1 |
|- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) ) |
12 |
|
zmodfzo |
|- ( ( ( I - N ) e. ZZ /\ ( # ` W ) e. NN ) -> ( ( I - N ) mod ( # ` W ) ) e. ( 0 ..^ ( # ` W ) ) ) |
13 |
11 12
|
syl |
|- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( I - N ) mod ( # ` W ) ) e. ( 0 ..^ ( # ` W ) ) ) |
14 |
|
cshwidxmod |
|- ( ( W e. Word V /\ N e. ZZ /\ ( ( I - N ) mod ( # ` W ) ) e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) ) |
15 |
13 14
|
syld3an3 |
|- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) ) |
16 |
|
elfzoelz |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> I e. ZZ ) |
17 |
16
|
adantl |
|- ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) -> I e. ZZ ) |
18 |
17 4
|
sylan |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( I - N ) e. ZZ ) |
19 |
18
|
zred |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( I - N ) e. RR ) |
20 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
21 |
20
|
adantl |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> N e. RR ) |
22 |
|
nnrp |
|- ( ( # ` W ) e. NN -> ( # ` W ) e. RR+ ) |
23 |
22
|
ad3antlr |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( # ` W ) e. RR+ ) |
24 |
|
modaddmod |
|- ( ( ( I - N ) e. RR /\ N e. RR /\ ( # ` W ) e. RR+ ) -> ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) = ( ( ( I - N ) + N ) mod ( # ` W ) ) ) |
25 |
19 21 23 24
|
syl3anc |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) = ( ( ( I - N ) + N ) mod ( # ` W ) ) ) |
26 |
|
nn0cn |
|- ( I e. NN0 -> I e. CC ) |
27 |
26
|
ad2antrr |
|- ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) -> I e. CC ) |
28 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
29 |
|
npcan |
|- ( ( I e. CC /\ N e. CC ) -> ( ( I - N ) + N ) = I ) |
30 |
27 28 29
|
syl2an |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( I - N ) + N ) = I ) |
31 |
30
|
oveq1d |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( ( I - N ) + N ) mod ( # ` W ) ) = ( I mod ( # ` W ) ) ) |
32 |
|
zmodidfzoimp |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> ( I mod ( # ` W ) ) = I ) |
33 |
32
|
ad2antlr |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( I mod ( # ` W ) ) = I ) |
34 |
25 31 33
|
3eqtrd |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) = I ) |
35 |
34
|
fveq2d |
|- ( ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) /\ N e. ZZ ) -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) |
36 |
35
|
ex |
|- ( ( ( I e. NN0 /\ ( # ` W ) e. NN ) /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) |
37 |
36
|
ex |
|- ( ( I e. NN0 /\ ( # ` W ) e. NN ) -> ( I e. ( 0 ..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) ) |
38 |
37
|
3adant3 |
|- ( ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) -> ( I e. ( 0 ..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) ) |
39 |
1 38
|
sylbi |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> ( I e. ( 0 ..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) ) |
40 |
39
|
pm2.43i |
|- ( I e. ( 0 ..^ ( # ` W ) ) -> ( N e. ZZ -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) ) |
41 |
40
|
impcom |
|- ( ( N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) |
42 |
41
|
3adant1 |
|- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` ( ( ( ( I - N ) mod ( # ` W ) ) + N ) mod ( # ` W ) ) ) = ( W ` I ) ) |
43 |
15 42
|
eqtrd |
|- ( ( W e. Word V /\ N e. ZZ /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( I - N ) mod ( # ` W ) ) ) = ( W ` I ) ) |