Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> i e. ( 0 ..^ ( N - M ) ) ) |
2 |
|
fzssz |
|- ( 0 ... ( # ` W ) ) C_ ZZ |
3 |
|
simpl3 |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> N e. ( 0 ... ( # ` W ) ) ) |
4 |
2 3
|
sseldi |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> N e. ZZ ) |
5 |
|
fzssz |
|- ( 0 ... N ) C_ ZZ |
6 |
|
simpl2 |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> M e. ( 0 ... N ) ) |
7 |
5 6
|
sseldi |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> M e. ZZ ) |
8 |
|
fzoaddel2 |
|- ( ( i e. ( 0 ..^ ( N - M ) ) /\ N e. ZZ /\ M e. ZZ ) -> ( i + M ) e. ( M ..^ N ) ) |
9 |
1 4 7 8
|
syl3anc |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ i e. ( 0 ..^ ( N - M ) ) ) -> ( i + M ) e. ( M ..^ N ) ) |
10 |
|
simpr |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> j e. ( M ..^ N ) ) |
11 |
|
simpl2 |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> M e. ( 0 ... N ) ) |
12 |
5 11
|
sseldi |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> M e. ZZ ) |
13 |
12
|
zcnd |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> M e. CC ) |
14 |
|
simpl3 |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> N e. ( 0 ... ( # ` W ) ) ) |
15 |
2 14
|
sseldi |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> N e. ZZ ) |
16 |
15
|
zcnd |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> N e. CC ) |
17 |
13 16
|
pncan3d |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> ( M + ( N - M ) ) = N ) |
18 |
17
|
oveq2d |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> ( M ..^ ( M + ( N - M ) ) ) = ( M ..^ N ) ) |
19 |
10 18
|
eleqtrrd |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> j e. ( M ..^ ( M + ( N - M ) ) ) ) |
20 |
15 12
|
zsubcld |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> ( N - M ) e. ZZ ) |
21 |
|
fzosubel3 |
|- ( ( j e. ( M ..^ ( M + ( N - M ) ) ) /\ ( N - M ) e. ZZ ) -> ( j - M ) e. ( 0 ..^ ( N - M ) ) ) |
22 |
19 20 21
|
syl2anc |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> ( j - M ) e. ( 0 ..^ ( N - M ) ) ) |
23 |
|
simpr |
|- ( ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) /\ i = ( j - M ) ) -> i = ( j - M ) ) |
24 |
23
|
oveq1d |
|- ( ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) /\ i = ( j - M ) ) -> ( i + M ) = ( ( j - M ) + M ) ) |
25 |
24
|
eqeq2d |
|- ( ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) /\ i = ( j - M ) ) -> ( j = ( i + M ) <-> j = ( ( j - M ) + M ) ) ) |
26 |
|
fzossz |
|- ( M ..^ N ) C_ ZZ |
27 |
26 10
|
sseldi |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> j e. ZZ ) |
28 |
27
|
zcnd |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> j e. CC ) |
29 |
28 13
|
npcand |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> ( ( j - M ) + M ) = j ) |
30 |
29
|
eqcomd |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> j = ( ( j - M ) + M ) ) |
31 |
22 25 30
|
rspcedvd |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j e. ( M ..^ N ) ) -> E. i e. ( 0 ..^ ( N - M ) ) j = ( i + M ) ) |
32 |
|
eqcom |
|- ( y = ( W ` j ) <-> ( W ` j ) = y ) |
33 |
|
simpr |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j = ( i + M ) ) -> j = ( i + M ) ) |
34 |
33
|
fveq2d |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j = ( i + M ) ) -> ( W ` j ) = ( W ` ( i + M ) ) ) |
35 |
34
|
eqeq2d |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j = ( i + M ) ) -> ( y = ( W ` j ) <-> y = ( W ` ( i + M ) ) ) ) |
36 |
32 35
|
bitr3id |
|- ( ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) /\ j = ( i + M ) ) -> ( ( W ` j ) = y <-> y = ( W ` ( i + M ) ) ) ) |
37 |
9 31 36
|
rexxfrd |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( E. j e. ( M ..^ N ) ( W ` j ) = y <-> E. i e. ( 0 ..^ ( N - M ) ) y = ( W ` ( i + M ) ) ) ) |
38 |
|
eqid |
|- ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) = ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) |
39 |
|
fvex |
|- ( W ` ( i + M ) ) e. _V |
40 |
38 39
|
elrnmpti |
|- ( y e. ran ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) <-> E. i e. ( 0 ..^ ( N - M ) ) y = ( W ` ( i + M ) ) ) |
41 |
37 40
|
bitr4di |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( E. j e. ( M ..^ N ) ( W ` j ) = y <-> y e. ran ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) ) ) |
42 |
|
wrdf |
|- ( W e. Word V -> W : ( 0 ..^ ( # ` W ) ) --> V ) |
43 |
42
|
3ad2ant1 |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> W : ( 0 ..^ ( # ` W ) ) --> V ) |
44 |
43
|
ffnd |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> W Fn ( 0 ..^ ( # ` W ) ) ) |
45 |
|
elfzuz |
|- ( M e. ( 0 ... N ) -> M e. ( ZZ>= ` 0 ) ) |
46 |
45
|
3ad2ant2 |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> M e. ( ZZ>= ` 0 ) ) |
47 |
|
fzoss1 |
|- ( M e. ( ZZ>= ` 0 ) -> ( M ..^ N ) C_ ( 0 ..^ N ) ) |
48 |
46 47
|
syl |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( M ..^ N ) C_ ( 0 ..^ N ) ) |
49 |
|
elfzuz3 |
|- ( N e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` N ) ) |
50 |
49
|
3ad2ant3 |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( # ` W ) e. ( ZZ>= ` N ) ) |
51 |
|
fzoss2 |
|- ( ( # ` W ) e. ( ZZ>= ` N ) -> ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
52 |
50 51
|
syl |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
53 |
48 52
|
sstrd |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( M ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
54 |
44 53
|
fvelimabd |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( y e. ( W " ( M ..^ N ) ) <-> E. j e. ( M ..^ N ) ( W ` j ) = y ) ) |
55 |
|
swrdval2 |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( W substr <. M , N >. ) = ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) ) |
56 |
55
|
rneqd |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ran ( W substr <. M , N >. ) = ran ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) ) |
57 |
56
|
eleq2d |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( y e. ran ( W substr <. M , N >. ) <-> y e. ran ( i e. ( 0 ..^ ( N - M ) ) |-> ( W ` ( i + M ) ) ) ) ) |
58 |
41 54 57
|
3bitr4rd |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ( y e. ran ( W substr <. M , N >. ) <-> y e. ( W " ( M ..^ N ) ) ) ) |
59 |
58
|
eqrdv |
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ran ( W substr <. M , N >. ) = ( W " ( M ..^ N ) ) ) |