Step |
Hyp |
Ref |
Expression |
1 |
|
swrdf1.w |
|- ( ph -> W e. Word D ) |
2 |
|
swrdf1.m |
|- ( ph -> M e. ( 0 ... N ) ) |
3 |
|
swrdf1.n |
|- ( ph -> N e. ( 0 ... ( # ` W ) ) ) |
4 |
|
swrdf1.1 |
|- ( ph -> W : dom W -1-1-> D ) |
5 |
|
swrdf |
|- ( ( W e. Word D /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. M , N >. ) : ( 0 ..^ ( N - M ) ) --> D ) |
6 |
1 2 3 5
|
syl3anc |
|- ( ph -> ( W substr <. M , N >. ) : ( 0 ..^ ( N - M ) ) --> D ) |
7 |
6
|
ffdmd |
|- ( ph -> ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) --> D ) |
8 |
|
fzossz |
|- ( 0 ..^ ( N - M ) ) C_ ZZ |
9 |
|
simpllr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> i e. dom ( W substr <. M , N >. ) ) |
10 |
6
|
fdmd |
|- ( ph -> dom ( W substr <. M , N >. ) = ( 0 ..^ ( N - M ) ) ) |
11 |
10
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> dom ( W substr <. M , N >. ) = ( 0 ..^ ( N - M ) ) ) |
12 |
9 11
|
eleqtrd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> i e. ( 0 ..^ ( N - M ) ) ) |
13 |
8 12
|
sselid |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> i e. ZZ ) |
14 |
13
|
zcnd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> i e. CC ) |
15 |
|
simplr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> j e. dom ( W substr <. M , N >. ) ) |
16 |
15 11
|
eleqtrd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> j e. ( 0 ..^ ( N - M ) ) ) |
17 |
8 16
|
sselid |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> j e. ZZ ) |
18 |
17
|
zcnd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> j e. CC ) |
19 |
2
|
elfzelzd |
|- ( ph -> M e. ZZ ) |
20 |
19
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> M e. ZZ ) |
21 |
20
|
zcnd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> M e. CC ) |
22 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> W : dom W -1-1-> D ) |
23 |
|
elfzuz |
|- ( M e. ( 0 ... N ) -> M e. ( ZZ>= ` 0 ) ) |
24 |
|
fzoss1 |
|- ( M e. ( ZZ>= ` 0 ) -> ( M ..^ N ) C_ ( 0 ..^ N ) ) |
25 |
2 23 24
|
3syl |
|- ( ph -> ( M ..^ N ) C_ ( 0 ..^ N ) ) |
26 |
|
elfzuz3 |
|- ( N e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` N ) ) |
27 |
|
fzoss2 |
|- ( ( # ` W ) e. ( ZZ>= ` N ) -> ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
28 |
3 26 27
|
3syl |
|- ( ph -> ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
29 |
25 28
|
sstrd |
|- ( ph -> ( M ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
30 |
29
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( M ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
31 |
3
|
elfzelzd |
|- ( ph -> N e. ZZ ) |
32 |
31
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> N e. ZZ ) |
33 |
|
fzoaddel2 |
|- ( ( i e. ( 0 ..^ ( N - M ) ) /\ N e. ZZ /\ M e. ZZ ) -> ( i + M ) e. ( M ..^ N ) ) |
34 |
12 32 20 33
|
syl3anc |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( i + M ) e. ( M ..^ N ) ) |
35 |
30 34
|
sseldd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( i + M ) e. ( 0 ..^ ( # ` W ) ) ) |
36 |
|
wrddm |
|- ( W e. Word D -> dom W = ( 0 ..^ ( # ` W ) ) ) |
37 |
1 36
|
syl |
|- ( ph -> dom W = ( 0 ..^ ( # ` W ) ) ) |
38 |
37
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> dom W = ( 0 ..^ ( # ` W ) ) ) |
39 |
35 38
|
eleqtrrd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( i + M ) e. dom W ) |
40 |
|
fzoaddel2 |
|- ( ( j e. ( 0 ..^ ( N - M ) ) /\ N e. ZZ /\ M e. ZZ ) -> ( j + M ) e. ( M ..^ N ) ) |
41 |
16 32 20 40
|
syl3anc |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( j + M ) e. ( M ..^ N ) ) |
42 |
30 41
|
sseldd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( j + M ) e. ( 0 ..^ ( # ` W ) ) ) |
43 |
42 38
|
eleqtrrd |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( j + M ) e. dom W ) |
44 |
|
simpr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) |
45 |
1
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> W e. Word D ) |
46 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> M e. ( 0 ... N ) ) |
47 |
3
|
ad3antrrr |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> N e. ( 0 ... ( # ` W ) ) ) |
48 |
|
swrdfv |
|- ( ( ( W e. Word D /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> ( ( W substr <. M , N >. ) ` i ) = ( W ` ( i + M ) ) ) |
49 |
45 46 47 12 48
|
syl31anc |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( ( W substr <. M , N >. ) ` i ) = ( W ` ( i + M ) ) ) |
50 |
|
swrdfv |
|- ( ( ( W e. Word D /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( 0 ..^ ( N - M ) ) ) -> ( ( W substr <. M , N >. ) ` j ) = ( W ` ( j + M ) ) ) |
51 |
45 46 47 16 50
|
syl31anc |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( ( W substr <. M , N >. ) ` j ) = ( W ` ( j + M ) ) ) |
52 |
44 49 51
|
3eqtr3d |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( W ` ( i + M ) ) = ( W ` ( j + M ) ) ) |
53 |
|
f1veqaeq |
|- ( ( W : dom W -1-1-> D /\ ( ( i + M ) e. dom W /\ ( j + M ) e. dom W ) ) -> ( ( W ` ( i + M ) ) = ( W ` ( j + M ) ) -> ( i + M ) = ( j + M ) ) ) |
54 |
53
|
anassrs |
|- ( ( ( W : dom W -1-1-> D /\ ( i + M ) e. dom W ) /\ ( j + M ) e. dom W ) -> ( ( W ` ( i + M ) ) = ( W ` ( j + M ) ) -> ( i + M ) = ( j + M ) ) ) |
55 |
54
|
imp |
|- ( ( ( ( W : dom W -1-1-> D /\ ( i + M ) e. dom W ) /\ ( j + M ) e. dom W ) /\ ( W ` ( i + M ) ) = ( W ` ( j + M ) ) ) -> ( i + M ) = ( j + M ) ) |
56 |
22 39 43 52 55
|
syl1111anc |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> ( i + M ) = ( j + M ) ) |
57 |
14 18 21 56
|
addcan2ad |
|- ( ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) /\ ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) ) -> i = j ) |
58 |
57
|
ex |
|- ( ( ( ph /\ i e. dom ( W substr <. M , N >. ) ) /\ j e. dom ( W substr <. M , N >. ) ) -> ( ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) -> i = j ) ) |
59 |
58
|
anasss |
|- ( ( ph /\ ( i e. dom ( W substr <. M , N >. ) /\ j e. dom ( W substr <. M , N >. ) ) ) -> ( ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) -> i = j ) ) |
60 |
59
|
ralrimivva |
|- ( ph -> A. i e. dom ( W substr <. M , N >. ) A. j e. dom ( W substr <. M , N >. ) ( ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) -> i = j ) ) |
61 |
|
dff13 |
|- ( ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) -1-1-> D <-> ( ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) --> D /\ A. i e. dom ( W substr <. M , N >. ) A. j e. dom ( W substr <. M , N >. ) ( ( ( W substr <. M , N >. ) ` i ) = ( ( W substr <. M , N >. ) ` j ) -> i = j ) ) ) |
62 |
7 60 61
|
sylanbrc |
|- ( ph -> ( W substr <. M , N >. ) : dom ( W substr <. M , N >. ) -1-1-> D ) |