Metamath Proof Explorer

Theorem swrdrn3

Description: Express the range of a subword. Stronger version of swrdrn2 . (Contributed by Thierry Arnoux, 13-Dec-2023)

Ref Expression
Assertion swrdrn3 
|- ( ( W e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` W ) ) ) -> ran ( W substr <. M , N >. ) = ( W " ( M ..^ N ) ) )

Proof

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