Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> W e. Word V ) |
2 |
|
elfzoelz |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> N e. ZZ ) |
3 |
2
|
adantl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> N e. ZZ ) |
4 |
|
ubmelm1fzo |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( ( # ` W ) - N ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
5 |
4
|
adantl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - N ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
6 |
|
cshwidxmod |
|- ( ( W e. Word V /\ N e. ZZ /\ ( ( ( # ` W ) - N ) - 1 ) e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( ( # ` W ) - N ) - 1 ) ) = ( W ` ( ( ( ( ( # ` W ) - N ) - 1 ) + N ) mod ( # ` W ) ) ) ) |
7 |
1 3 5 6
|
syl3anc |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( ( # ` W ) - N ) - 1 ) ) = ( W ` ( ( ( ( ( # ` W ) - N ) - 1 ) + N ) mod ( # ` W ) ) ) ) |
8 |
|
elfzoel2 |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( # ` W ) e. ZZ ) |
9 |
8
|
zcnd |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( # ` W ) e. CC ) |
10 |
2
|
zcnd |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> N e. CC ) |
11 |
|
1cnd |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> 1 e. CC ) |
12 |
|
nnpcan |
|- ( ( ( # ` W ) e. CC /\ N e. CC /\ 1 e. CC ) -> ( ( ( ( # ` W ) - N ) - 1 ) + N ) = ( ( # ` W ) - 1 ) ) |
13 |
9 10 11 12
|
syl3anc |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( ( ( # ` W ) - N ) - 1 ) + N ) = ( ( # ` W ) - 1 ) ) |
14 |
13
|
oveq1d |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( ( ( ( # ` W ) - N ) - 1 ) + N ) mod ( # ` W ) ) = ( ( ( # ` W ) - 1 ) mod ( # ` W ) ) ) |
15 |
14
|
adantl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( ( ( # ` W ) - N ) - 1 ) + N ) mod ( # ` W ) ) = ( ( ( # ` W ) - 1 ) mod ( # ` W ) ) ) |
16 |
|
elfzo0 |
|- ( N e. ( 0 ..^ ( # ` W ) ) <-> ( N e. NN0 /\ ( # ` W ) e. NN /\ N < ( # ` W ) ) ) |
17 |
|
nnre |
|- ( ( # ` W ) e. NN -> ( # ` W ) e. RR ) |
18 |
|
peano2rem |
|- ( ( # ` W ) e. RR -> ( ( # ` W ) - 1 ) e. RR ) |
19 |
17 18
|
syl |
|- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. RR ) |
20 |
|
nnrp |
|- ( ( # ` W ) e. NN -> ( # ` W ) e. RR+ ) |
21 |
19 20
|
jca |
|- ( ( # ` W ) e. NN -> ( ( ( # ` W ) - 1 ) e. RR /\ ( # ` W ) e. RR+ ) ) |
22 |
21
|
3ad2ant2 |
|- ( ( N e. NN0 /\ ( # ` W ) e. NN /\ N < ( # ` W ) ) -> ( ( ( # ` W ) - 1 ) e. RR /\ ( # ` W ) e. RR+ ) ) |
23 |
16 22
|
sylbi |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( ( # ` W ) - 1 ) e. RR /\ ( # ` W ) e. RR+ ) ) |
24 |
|
nnm1ge0 |
|- ( ( # ` W ) e. NN -> 0 <_ ( ( # ` W ) - 1 ) ) |
25 |
24
|
3ad2ant2 |
|- ( ( N e. NN0 /\ ( # ` W ) e. NN /\ N < ( # ` W ) ) -> 0 <_ ( ( # ` W ) - 1 ) ) |
26 |
16 25
|
sylbi |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> 0 <_ ( ( # ` W ) - 1 ) ) |
27 |
|
zre |
|- ( ( # ` W ) e. ZZ -> ( # ` W ) e. RR ) |
28 |
27
|
ltm1d |
|- ( ( # ` W ) e. ZZ -> ( ( # ` W ) - 1 ) < ( # ` W ) ) |
29 |
8 28
|
syl |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( # ` W ) - 1 ) < ( # ` W ) ) |
30 |
23 26 29
|
jca32 |
|- ( N e. ( 0 ..^ ( # ` W ) ) -> ( ( ( ( # ` W ) - 1 ) e. RR /\ ( # ` W ) e. RR+ ) /\ ( 0 <_ ( ( # ` W ) - 1 ) /\ ( ( # ` W ) - 1 ) < ( # ` W ) ) ) ) |
31 |
30
|
adantl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( ( # ` W ) - 1 ) e. RR /\ ( # ` W ) e. RR+ ) /\ ( 0 <_ ( ( # ` W ) - 1 ) /\ ( ( # ` W ) - 1 ) < ( # ` W ) ) ) ) |
32 |
|
modid |
|- ( ( ( ( ( # ` W ) - 1 ) e. RR /\ ( # ` W ) e. RR+ ) /\ ( 0 <_ ( ( # ` W ) - 1 ) /\ ( ( # ` W ) - 1 ) < ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) mod ( # ` W ) ) = ( ( # ` W ) - 1 ) ) |
33 |
31 32
|
syl |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( # ` W ) - 1 ) mod ( # ` W ) ) = ( ( # ` W ) - 1 ) ) |
34 |
15 33
|
eqtrd |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( ( ( ( # ` W ) - N ) - 1 ) + N ) mod ( # ` W ) ) = ( ( # ` W ) - 1 ) ) |
35 |
34
|
fveq2d |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( W ` ( ( ( ( ( # ` W ) - N ) - 1 ) + N ) mod ( # ` W ) ) ) = ( W ` ( ( # ` W ) - 1 ) ) ) |
36 |
7 35
|
eqtrd |
|- ( ( W e. Word V /\ N e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` ( ( ( # ` W ) - N ) - 1 ) ) = ( W ` ( ( # ` W ) - 1 ) ) ) |