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Theorem repswcshw 12567
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 12551 . . . . 5  |-  ( (/) cyclShift  I )  =  (/)
2 repsw0 12536 . . . . . 6  |-  ( S  e.  V  ->  ( S repeatS  0 )  =  (/) )
32oveq1d 6218 . . . . 5  |-  ( S  e.  V  ->  (
( S repeatS  0 ) cyclShift  I )  =  (
(/) cyclShift  I ) )
41, 3, 23eqtr4a 2521 . . . 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 6211 . . . . 5  |-  ( N  =  0  ->  ( S repeatS  N )  =  ( S repeatS  0 ) )
76oveq1d 6218 . . . 4  |-  ( N  =  0  ->  (
( S repeatS  N ) cyclShift  I )  =  ( ( S repeatS  0 ) cyclShift  I
) )
87, 6eqeq12d 2476 . . 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 2650 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
12 elnnne0 10707 . . . . . 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 1297 . . 3  |-  ( -.  N  =  0  -> 
( ( S  e.  V  /\  N  e. 
NN0  /\  I  e.  ZZ )  ->  ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ ) ) )
17 nnnn0 10700 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
1817anim2i 569 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S  e.  V  /\  N  e.  NN0 ) )
19 repsw 12534 . . . . . . 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 12549 . . . . 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 12535 . . . . . . . . . 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 6219 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( I  mod  ( # `
 ( S repeatS  N
) ) )  =  ( I  mod  N
) )
2827, 26opeq12d 4178 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  -> 
<. ( I  mod  ( # `
 ( S repeatS  N
) ) ) ,  ( # `  ( S repeatS  N ) ) >.  =  <. ( I  mod  N ) ,  N >. )
2928oveq2d 6219 . . . . . 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 4177 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  -> 
<. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >.  =  <. 0 ,  ( I  mod  N ) >. )
3130oveq2d 6219 . . . . . 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 6221 . . . . 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 11847 . . . . . . . . . 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 10443 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
4140leidd 10020 . . . . . . . 8  |-  ( N  e.  NN  ->  N  <_  N )
42413ad2ant2 1010 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  N  <_  N )
43 repswswrd 12543 . . . . . . 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 1219 . . . . . 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 10708 . . . . . . . . 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 10764 . . . . . . . . . 10  |-  ( I  e.  ZZ  ->  I  e.  RR )
49 nnrp 11114 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
50 modcl 11832 . . . . . . . . . 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 11838 . . . . . . . . . 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 9636 . . . . . . . 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 12543 . . . . . . 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 1219 . . . . . 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 6221 . . . . 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 10751 . . . . . . . . . 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 9636 . . . . . . . 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 10744 . . . . . . . 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 10753 . . . . . . . . 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 10744 . . . . . . . 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 12544 . . . . . 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 1219 . . . . 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 10444 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
8079adantl 466 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
8135nn0cnd 10752 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  CC )
82 0cnd 9493 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  0  e.  CC )
8380, 81, 82npncand 9857 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  ( N  - 
0 ) )
8479subid1d 9822 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  0 )  =  N )
8584adantl 466 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( N  -  0 )  =  N )
8683, 85eqtrd 2495 . . . . . . . 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 6219 . . . . 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 2499 . . . 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 2499 . . 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 1370    e. wcel 1758    =/= wne 2648   (/)c0 3748   <.cop 3994   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9394   RRcr 9395   0cc0 9396    + caddc 9399    < clt 9532    <_ cle 9533    - cmin 9709   NNcn 10436   NN0cn0 10693   ZZcz 10760   RR+crp 11105    mod cmo 11828   #chash 12223  Word cword 12342   concat cconcat 12344   substr csubstr 12346   repeatS creps 12349   cyclShift ccsh 12546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fzo 11669  df-fl 11762  df-mod 11829  df-hash 12224  df-word 12350  df-concat 12352  df-substr 12354  df-reps 12357  df-csh 12547
This theorem is referenced by:  cshwrepswhash1  14250
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