MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  repswcshw Structured version   Unicode version

Theorem repswcshw 12836
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 12820 . . . . 5  |-  ( (/) cyclShift  I )  =  (/)
2 repsw0 12805 . . . . . 6  |-  ( S  e.  V  ->  ( S repeatS  0 )  =  (/) )
32oveq1d 6293 . . . . 5  |-  ( S  e.  V  ->  (
( S repeatS  0 ) cyclShift  I )  =  (
(/) cyclShift  I ) )
41, 3, 23eqtr4a 2469 . . . 4  |-  ( S  e.  V  ->  (
( S repeatS  0 ) cyclShift  I )  =  ( S repeatS  0 ) )
543ad2ant1 1018 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN0  /\  I  e.  ZZ )  ->  (
( S repeatS  0 ) cyclShift  I )  =  ( S repeatS  0 ) )
6 oveq2 6286 . . . . 5  |-  ( N  =  0  ->  ( S repeatS  N )  =  ( S repeatS  0 ) )
76oveq1d 6293 . . . 4  |-  ( N  =  0  ->  (
( S repeatS  N ) cyclShift  I )  =  ( ( S repeatS  0 ) cyclShift  I
) )
87, 6eqeq12d 2424 . . 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 2600 . . . . 5  |-  ( N  =/=  0  <->  -.  N  =  0 )
12 elnnne0 10850 . . . . . 6  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
1312simplbi2com 625 . . . . 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 1308 . . 3  |-  ( -.  N  =  0  -> 
( ( S  e.  V  /\  N  e. 
NN0  /\  I  e.  ZZ )  ->  ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ ) ) )
17 nnnn0 10843 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  NN0 )
1817anim2i 567 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S  e.  V  /\  N  e.  NN0 ) )
19 repsw 12803 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( S repeatS  N )  e. Word  V )
2018, 19syl 17 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( S repeatS  N )  e. Word  V )
21 cshword 12818 . . . . 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 ) ) >.
) ++  ( ( S repeatS  N ) substr  <. 0 ,  ( I  mod  ( # `
 ( S repeatS  N
) ) ) >.
) ) )
2220, 21stoic3 1630 . . . 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 ) ) >.
) ++  ( ( S repeatS  N ) substr  <. 0 ,  ( I  mod  ( # `
 ( S repeatS  N
) ) ) >.
) ) )
23 repswlen 12804 . . . . . . . . . 10  |-  ( ( S  e.  V  /\  N  e.  NN0 )  -> 
( # `  ( S repeatS  N ) )  =  N )
2418, 23syl 17 . . . . . . . . 9  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( # `  ( S repeatS  N ) )  =  N )
2524oveq2d 6294 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( I  mod  ( # `
 ( S repeatS  N
) ) )  =  ( I  mod  N
) )
2625, 24opeq12d 4167 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  -> 
<. ( I  mod  ( # `
 ( S repeatS  N
) ) ) ,  ( # `  ( S repeatS  N ) ) >.  =  <. ( I  mod  N ) ,  N >. )
2726oveq2d 6294 . . . . . 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 >. ) )
2825opeq2d 4166 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN )  -> 
<. 0 ,  ( I  mod  ( # `  ( S repeatS  N )
) ) >.  =  <. 0 ,  ( I  mod  N ) >. )
2928oveq2d 6294 . . . . . 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
) >. ) )
3027, 29oveq12d 6296 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN )  ->  ( ( ( S repeatS  N ) substr  <. ( I  mod  ( # `  ( S repeatS  N ) ) ) ,  ( # `  ( S repeatS  N ) ) >.
) ++  ( ( S repeatS  N ) substr  <. 0 ,  ( I  mod  ( # `
 ( S repeatS  N
) ) ) >.
) )  =  ( ( ( S repeatS  N
) substr  <. ( I  mod  N ) ,  N >. ) ++  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) ) )
31303adant3 1017 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( ( S repeatS  N
) substr  <. ( I  mod  ( # `  ( S repeatS  N ) ) ) ,  ( # `  ( S repeatS  N ) ) >.
) ++  ( ( S repeatS  N ) substr  <. 0 ,  ( I  mod  ( # `
 ( S repeatS  N
) ) ) >.
) )  =  ( ( ( S repeatS  N
) substr  <. ( I  mod  N ) ,  N >. ) ++  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) ) )
32183adant3 1017 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( S  e.  V  /\  N  e.  NN0 ) )
33 zmodcl 12054 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  NN0 )
3433ancoms 451 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  e.  NN0 )
3517adantr 463 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  N  e.  NN0 )
3634, 35jca 530 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( ( I  mod  N )  e.  NN0  /\  N  e.  NN0 ) )
37363adant1 1015 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( I  mod  N
)  e.  NN0  /\  N  e.  NN0 ) )
38 nnre 10583 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  RR )
3938leidd 10159 . . . . . . . 8  |-  ( N  e.  NN  ->  N  <_  N )
40393ad2ant2 1019 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  N  <_  N )
41 repswswrd 12812 . . . . . . 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 ) ) ) )
4232, 37, 40, 41syl3anc 1230 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) substr  <.
( I  mod  N
) ,  N >. )  =  ( S repeatS  ( N  -  ( I  mod  N ) ) ) )
43 0nn0 10851 . . . . . . . . 9  |-  0  e.  NN0
4434, 43jctil 535 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( 0  e.  NN0  /\  ( I  mod  N
)  e.  NN0 )
)
45443adant1 1015 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
0  e.  NN0  /\  ( I  mod  N )  e.  NN0 ) )
46 zre 10909 . . . . . . . . . 10  |-  ( I  e.  ZZ  ->  I  e.  RR )
47 nnrp 11274 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
48 modcl 12038 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  N  e.  RR+ )  -> 
( I  mod  N
)  e.  RR )
4946, 47, 48syl2anr 476 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  e.  RR )
5038adantr 463 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  N  e.  RR )
51 modlt 12045 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  N  e.  RR+ )  -> 
( I  mod  N
)  <  N )
5246, 47, 51syl2anr 476 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  <  N )
5349, 50, 52ltled 9765 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  <_  N )
54533adant1 1015 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
I  mod  N )  <_  N )
55 repswswrd 12812 . . . . . . 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 ) ) )
5632, 45, 54, 55syl3anc 1230 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) substr  <.
0 ,  ( I  mod  N ) >.
)  =  ( S repeatS 
( ( I  mod  N )  -  0 ) ) )
5742, 56oveq12d 6296 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( ( S repeatS  N
) substr  <. ( I  mod  N ) ,  N >. ) ++  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) )  =  ( ( S repeatS  ( N  -  ( I  mod  N ) ) ) ++  ( S repeatS  ( ( I  mod  N )  - 
0 ) ) ) )
58 simp1 997 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  S  e.  V )
5933nn0red 10894 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  RR )
6059ancoms 451 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  e.  RR )
6160, 50, 52ltled 9765 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( I  mod  N
)  <_  N )
62613adant1 1015 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
I  mod  N )  <_  N )
63343adant1 1015 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
I  mod  N )  e.  NN0 )
64173ad2ant2 1019 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  N  e.  NN0 )
65 nn0sub 10887 . . . . . . . 8  |-  ( ( ( I  mod  N
)  e.  NN0  /\  N  e.  NN0 )  -> 
( ( I  mod  N )  <_  N  <->  ( N  -  ( I  mod  N ) )  e.  NN0 ) )
6663, 64, 65syl2anc 659 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( I  mod  N
)  <_  N  <->  ( N  -  ( I  mod  N ) )  e.  NN0 ) )
6762, 66mpbid 210 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( N  -  ( I  mod  N ) )  e. 
NN0 )
6833nn0ge0d 10896 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  0  <_  ( I  mod  N ) )
6968ancoms 451 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  0  <_  ( I  mod  N ) )
70693adant1 1015 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  0  <_  ( I  mod  N
) )
7163, 43jctil 535 . . . . . . . 8  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
0  e.  NN0  /\  ( I  mod  N )  e.  NN0 ) )
72 nn0sub 10887 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  ( I  mod  N )  e.  NN0 )  -> 
( 0  <_  (
I  mod  N )  <->  ( ( I  mod  N
)  -  0 )  e.  NN0 ) )
7371, 72syl 17 . . . . . . 7  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
0  <_  ( I  mod  N )  <->  ( (
I  mod  N )  -  0 )  e. 
NN0 ) )
7470, 73mpbid 210 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( I  mod  N
)  -  0 )  e.  NN0 )
75 repswccat 12813 . . . . . 6  |-  ( ( S  e.  V  /\  ( N  -  (
I  mod  N )
)  e.  NN0  /\  ( ( I  mod  N )  -  0 )  e.  NN0 )  -> 
( ( S repeatS  ( N  -  ( I  mod  N ) ) ) ++  ( S repeatS  ( (
I  mod  N )  -  0 ) ) )  =  ( S repeatS 
( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) ) ) )
7658, 67, 74, 75syl3anc 1230 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  ( N  -  ( I  mod  N ) ) ) ++  ( S repeatS  ( ( I  mod  N )  - 
0 ) ) )  =  ( S repeatS  (
( N  -  (
I  mod  N )
)  +  ( ( I  mod  N )  -  0 ) ) ) )
77 nncn 10584 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
7877adantl 464 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  N  e.  CC )
7933nn0cnd 10895 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( I  mod  N
)  e.  CC )
80 0cnd 9619 . . . . . . . . . 10  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  0  e.  CC )
8178, 79, 80npncand 9991 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  ( N  - 
0 ) )
8277subid1d 9956 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  -  0 )  =  N )
8382adantl 464 . . . . . . . . 9  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( N  -  0 )  =  N )
8481, 83eqtrd 2443 . . . . . . . 8  |-  ( ( I  e.  ZZ  /\  N  e.  NN )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  N )
8584ancoms 451 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ZZ )  ->  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) )  =  N )
86853adant1 1015 . . . . . 6  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( N  -  (
I  mod  N )
)  +  ( ( I  mod  N )  -  0 ) )  =  N )
8786oveq2d 6294 . . . . 5  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  ( S repeatS  ( ( N  -  ( I  mod  N ) )  +  ( ( I  mod  N )  -  0 ) ) )  =  ( S repeatS  N ) )
8857, 76, 873eqtrd 2447 . . . 4  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( ( S repeatS  N
) substr  <. ( I  mod  N ) ,  N >. ) ++  ( ( S repeatS  N
) substr  <. 0 ,  ( I  mod  N )
>. ) )  =  ( S repeatS  N ) )
8922, 31, 883eqtrd 2447 . . 3  |-  ( ( S  e.  V  /\  N  e.  NN  /\  I  e.  ZZ )  ->  (
( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N ) )
9016, 89syl6 31 . 2  |-  ( -.  N  =  0  -> 
( ( S  e.  V  /\  N  e. 
NN0  /\  I  e.  ZZ )  ->  ( ( S repeatS  N ) cyclShift  I )  =  ( S repeatS  N
) ) )
919, 90pm2.61i 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   (/)c0 3738   <.cop 3978   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522    + caddc 9525    < clt 9658    <_ cle 9659    - cmin 9841   NNcn 10576   NN0cn0 10836   ZZcz 10905   RR+crp 11265    mod cmo 12034   #chash 12452  Word cword 12583   ++ cconcat 12585   substr csubstr 12587   repeatS creps 12590   cyclShift ccsh 12815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-hash 12453  df-word 12591  df-concat 12593  df-substr 12595  df-reps 12598  df-csh 12816
This theorem is referenced by:  cshwrepswhash1  14796
  Copyright terms: Public domain W3C validator