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Theorem repswcshw 12438
Description: A cyclically shifted "repeated symbol word". (Contributed by Alexander van der Vekens, 7-Nov-2018.)
Assertion
Ref Expression
repswcshw  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N ) )

Proof of Theorem repswcshw
StepHypRef Expression
1 0csh0 12422 . . . . 5  |-  ( (/) cyclShift  I )  =  (/)
2 repsw0 12407 . . . . . 6  |-  ( S  e.  V  ->  ( S repeatS  0 )  =  (/) )
32oveq1d 6101 . . . . 5  |-  ( S  e.  V  ->  (
( S repeatS  0 ) cyclShift  I )  =  (
(/) cyclShift  I ) )
41, 3, 23eqtr4a 2496 . . . 4  |-  ( S  e.  V  ->  (
( S repeatS  0 ) cyclShift  I )  =  ( S repeatS  0 ) )
543ad2ant1 1009 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ZZ )  ->  (
( S repeatS  0 ) cyclShift  I )  =  ( S repeatS  0 ) )
6 oveq2 6094 . . . . 5  |-  ( N  =  0  ->  ( S repeatS  N )  =  ( S repeatS  0 ) )
76oveq1d 6101 . . . 4  |-  ( N  =  0  ->  (
( S repeatS  N ) cyclShift  I )  =  ( ( S repeatS  0 ) cyclShift  I
) )
87, 6eqeq12d 2452 . . 3  |-  ( N  =  0  ->  (
( ( S repeatS  N
) cyclShift  I )  =  ( S repeatS  N )  <->  ( ( S repeatS  0 ) cyclShift  I )  =  ( S repeatS  0
) ) )
95, 8syl5ibr 221 . 2  |-  ( N  =  0  ->  (
( S  e.  V  /\  N  e.  NN0  /\  I  e.  ZZ )  ->  ( ( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N )
) )
10 idd 24 . . . 4  |-  ( -.  N  =  0  -> 
( S  e.  V  ->  S  e.  V ) )
11 df-ne 2603 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
12 elnnne0 10585 . . . . . 6  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
1312simplbi2com 627 . . . . 5  |-  ( N  =/=  0  ->  ( N  e.  NN0  ->  N  e.  NN ) )
1411, 13sylbir 213 . . . 4  |-  ( -.  N  =  0  -> 
( N  e.  NN0  ->  N  e.  NN ) )
15 idd 24 . . . 4  |-  ( -.  N  =  0  -> 
( I  e.  ZZ  ->  I  e.  ZZ ) )
1610, 14, 153anim123d 1296 . . 3  |-  ( -.  N  =  0  -> 
( ( S  e.  V  /\  N  e. 
NN0  /\  I  e.  ZZ )  ->  ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ ) ) )
17 nnnn0 10578 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
1817anim2i 569 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S  e.  V  /\  N  e.  NN0 ) )
19 repsw 12405 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  e. Word  V )
2018, 19syl 16 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S repeatS  N )  e. Word  V )
21203adant3 1008 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( S repeatS  N )  e. Word  V
)
22 simp3 990 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  I  e.  ZZ )
23 cshword 12420 . . . . 5  |-  ( ( ( S repeatS  N )  e. Word  V  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) cyclShift  I )  =  ( ( ( S repeatS  N ) substr  <.
( I  mod  ( # `
 ( S repeatS  N
) ) ) ,  ( # `  ( S repeatS  N ) ) >.
) concat  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >. )
) )
2421, 22, 23syl2anc 661 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) cyclShift  I )  =  ( ( ( S repeatS  N ) substr  <.
( I  mod  ( # `
 ( S repeatS  N
) ) ) ,  ( # `  ( S repeatS  N ) ) >.
) concat  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >. )
) )
25 repswlen 12406 . . . . . . . . . 10  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
2618, 25syl 16 . . . . . . . . 9  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( # `  ( S repeatS  N ) )  =  N )
2726oveq2d 6102 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( I  mod  ( # `
 ( S repeatS  N
) ) )  =  ( I  mod  N
) )
2827, 26opeq12d 4062 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  -> 
<. ( I  mod  ( # `
 ( S repeatS  N
) ) ) ,  ( # `  ( S repeatS  N ) ) >.  =  <. ( I  mod  N ) ,  N >. )
2928oveq2d 6102 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( ( S repeatS  N
) substr  <. ( I  mod  ( # `  ( S repeatS  N ) ) ) ,  ( # `  ( S repeatS  N ) ) >.
)  =  ( ( S repeatS  N ) substr  <. (
I  mod  N ) ,  N >. ) )
3027opeq2d 4061 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  -> 
<. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >.  =  <. 0 ,  ( I  mod  N ) >. )
3130oveq2d 6102 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >. )  =  ( ( S repeatS  N ) substr  <. 0 ,  ( I  mod  N
) >. ) )
3229, 31oveq12d 6104 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( ( ( S repeatS  N ) substr  <. ( I  mod  ( # `  ( S repeatS  N ) ) ) ,  ( # `  ( S repeatS  N ) ) >.
) concat  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >. )
)  =  ( ( ( S repeatS  N ) substr  <.
( I  mod  N
) ,  N >. ) concat 
( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) ) )
33323adant3 1008 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( ( S repeatS  N
) substr  <. ( I  mod  ( # `  ( S repeatS  N ) ) ) ,  ( # `  ( S repeatS  N ) ) >.
) concat  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >. )
)  =  ( ( ( S repeatS  N ) substr  <.
( I  mod  N
) ,  N >. ) concat 
( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) ) )
34183adant3 1008 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( S  e.  V  /\  N  e.  NN0 ) )
35 zmodcl 11719 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  NN0 )
3635ancoms 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  e.  NN0 )
3717adantr 465 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  N  e.  NN0 )
3836, 37jca 532 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( ( I  mod  N )  e.  NN0  /\  N  e.  NN0 ) )
39383adant1 1006 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( I  mod  N
)  e.  NN0  /\  N  e.  NN0 ) )
40 nnre 10321 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
4140leidd 9898 . . . . . . . 8  |-  ( N  e.  NN  ->  N  <_  N )
42413ad2ant2 1010 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  N  <_  N )
43 repswswrd 12414 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  ( ( I  mod  N )  e. 
NN0  /\  N  e.  NN0 )  /\  N  <_  N )  ->  (
( S repeatS  N ) substr  <.
( I  mod  N
) ,  N >. )  =  ( S repeatS  ( N  -  ( I  mod  N ) ) ) )
4434, 39, 42, 43syl3anc 1218 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) substr  <.
( I  mod  N
) ,  N >. )  =  ( S repeatS  ( N  -  ( I  mod  N ) ) ) )
45 0nn0 10586 . . . . . . . . 9  |-  0  e.  NN0
4636, 45jctil 537 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( 0  e.  NN0  /\  ( I  mod  N
)  e.  NN0 )
)
47463adant1 1006 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
0  e.  NN0  /\  ( I  mod  N )  e.  NN0 ) )
48 zre 10642 . . . . . . . . . 10  |-  ( I  e.  ZZ  ->  I  e.  RR )
49 nnrp 10992 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
50 modcl 11704 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  N  e.  RR+ )  -> 
( I  mod  N
)  e.  RR )
5148, 49, 50syl2anr 478 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  e.  RR )
5240adantr 465 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  N  e.  RR )
53 modlt 11710 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  N  e.  RR+ )  -> 
( I  mod  N
)  <  N )
5448, 49, 53syl2anr 478 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  <  N )
5551, 52, 54ltled 9514 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  <_  N )
56553adant1 1006 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
I  mod  N )  <_  N )
57 repswswrd 12414 . . . . . . 7  |-  ( ( ( S  e.  V  /\  N  e.  NN0 )  /\  ( 0  e. 
NN0  /\  ( I  mod  N )  e.  NN0 )  /\  ( I  mod  N )  <_  N )  ->  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. )  =  ( S repeatS  ( ( I  mod  N )  -  0 ) ) )
5834, 47, 56, 57syl3anc 1218 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) substr  <.
0 ,  ( I  mod  N ) >.
)  =  ( S repeatS 
( ( I  mod  N )  -  0 ) ) )
5944, 58oveq12d 6104 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( ( S repeatS  N
) substr  <. ( I  mod  N ) ,  N >. ) concat 
( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) )  =  ( ( S repeatS  ( N  -  ( I  mod  N ) ) ) concat  ( S repeatS  ( ( I  mod  N )  -  0 ) ) ) )
60 simp1 988 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  S  e.  V )
6135nn0red 10629 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  RR )
6261ancoms 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  e.  RR )
6362, 52, 54ltled 9514 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  <_  N )
64633adant1 1006 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
I  mod  N )  <_  N )
65363adant1 1006 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
I  mod  N )  e.  NN0 )
66173ad2ant2 1010 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  N  e.  NN0 )
67 nn0sub 10622 . . . . . . . 8  |-  ( ( ( I  mod  N
)  e.  NN0  /\  N  e.  NN0 )  -> 
( ( I  mod  N )  <_  N  <->  ( N  -  ( I  mod  N ) )  e.  NN0 ) )
6865, 66, 67syl2anc 661 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( I  mod  N
)  <_  N  <->  ( N  -  ( I  mod  N ) )  e.  NN0 ) )
6964, 68mpbid 210 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( N  -  ( I  mod  N ) )  e. 
NN0 )
7035nn0ge0d 10631 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  0  <_  ( I  mod  N ) )
7170ancoms 453 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  0  <_  ( I  mod  N ) )
72713adant1 1006 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  0  <_  ( I  mod  N
) )
7365, 45jctil 537 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
0  e.  NN0  /\  ( I  mod  N )  e.  NN0 ) )
74 nn0sub 10622 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  ( I  mod  N )  e.  NN0 )  -> 
( 0  <_  (
I  mod  N )  <->  ( ( I  mod  N
)  -  0 )  e.  NN0 ) )
7573, 74syl 16 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
0  <_  ( I  mod  N )  <->  ( (
I  mod  N )  -  0 )  e. 
NN0 ) )
7672, 75mpbid 210 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( I  mod  N
)  -  0 )  e.  NN0 )
77 repswccat 12415 . . . . . 6  |-  ( ( S  e.  V  /\  ( N  -  (
I  mod  N )
)  e.  NN0  /\  ( ( I  mod  N )  -  0 )  e.  NN0 )  -> 
( ( S repeatS  ( N  -  ( I  mod  N ) ) ) concat 
( S repeatS  ( (
I  mod  N )  -  0 ) ) )  =  ( S repeatS 
( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) ) ) )
7860, 69, 76, 77syl3anc 1218 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  ( N  -  ( I  mod  N ) ) ) concat  ( S repeatS  ( ( I  mod  N )  -  0 ) ) )  =  ( S repeatS  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N
)  -  0 ) ) ) )
79 nncn 10322 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
8079adantl 466 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
8135nn0cnd 10630 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  CC )
82 0cnd 9371 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  0  e.  CC )
8380, 81, 82npncand 9735 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  ( N  - 
0 ) )
8479subid1d 9700 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  0 )  =  N )
8584adantl 466 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( N  -  0 )  =  N )
8683, 85eqtrd 2470 . . . . . . . 8  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  N )
8786ancoms 453 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  N )
88873adant1 1006 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( N  -  (
I  mod  N )
)  +  ( ( I  mod  N )  -  0 ) )  =  N )
8988oveq2d 6102 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( S repeatS  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) ) )  =  ( S repeatS  N ) )
9059, 78, 893eqtrd 2474 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( ( S repeatS  N
) substr  <. ( I  mod  N ) ,  N >. ) concat 
( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) )  =  ( S repeatS  N ) )
9124, 33, 903eqtrd 2474 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N ) )
9216, 91syl6 33 . 2  |-  ( -.  N  =  0  -> 
( ( S  e.  V  /\  N  e. 
NN0  /\  I  e.  ZZ )  ->  ( ( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N
) ) )
939, 92pm2.61i 164 1  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   (/)c0 3632   <.cop 3878   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274    + caddc 9277    < clt 9410    <_ cle 9411    - cmin 9587   NNcn 10314   NN0cn0 10571   ZZcz 10638   RR+crp 10983    mod cmo 11700   #chash 12095  Word cword 12213   concat cconcat 12215   substr csubstr 12217   repeatS creps 12220   cyclShift ccsh 12417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-hash 12096  df-word 12221  df-concat 12223  df-substr 12225  df-reps 12228  df-csh 12418
This theorem is referenced by:  cshwrepswhash1  14121
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