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Theorem cshw1 12801
Description: If cyclically shifting a word by 1 position results in the word itself, the word is build of identical symbols. Remark: also "valid" for an empty word! (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Proof shortened by AV, 1-Nov-2018.)
Assertion
Ref Expression
cshw1  |-  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
Distinct variable groups:    i, V    i, W

Proof of Theorem cshw1
StepHypRef Expression
1 ral0 3937 . . . 4  |-  A. i  e.  (/)  ( W `  i )  =  ( W `  0 )
2 oveq2 6304 . . . . . 6  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 0 ) )
3 fzo0 11847 . . . . . 6  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2514 . . . . 5  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  (/) )
54raleqdv 3060 . . . 4  |-  ( (
# `  W )  =  0  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  A. i  e.  (/)  ( W `  i )  =  ( W ` 
0 ) ) )
61, 5mpbiri 233 . . 3  |-  ( (
# `  W )  =  0  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
76a1d 25 . 2  |-  ( (
# `  W )  =  0  ->  (
( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
8 simprl 756 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  W  e. Word  V )
9 lencl 12568 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
10 1nn0 10832 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1110a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  e.  NN0 )
12 df-ne 2654 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  0  <->  -.  ( # `  W
)  =  0 )
13 elnnne0 10830 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  NN0  /\  ( # `  W
)  =/=  0 ) )
1413simplbi2com 627 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  0  ->  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  NN ) )
1512, 14sylbir 213 . . . . . . . . . . . . . . 15  |-  ( -.  ( # `  W
)  =  0  -> 
( ( # `  W
)  e.  NN0  ->  (
# `  W )  e.  NN ) )
1615adantr 465 . . . . . . . . . . . . . 14  |-  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  ( ( # `  W )  e.  NN0  ->  ( # `  W
)  e.  NN ) )
1716impcom 430 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  ( # `  W
)  e.  NN )
18 df-ne 2654 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  1  <->  -.  ( # `  W
)  =  1 )
1918biimpri 206 . . . . . . . . . . . . . . 15  |-  ( -.  ( # `  W
)  =  1  -> 
( # `  W )  =/=  1 )
2019ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  ( # `  W
)  =/=  1 )
21 nngt1ne1 10583 . . . . . . . . . . . . . . 15  |-  ( (
# `  W )  e.  NN  ->  ( 1  <  ( # `  W
)  <->  ( # `  W
)  =/=  1 ) )
2217, 21syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  ( 1  <  ( # `  W
)  <->  ( # `  W
)  =/=  1 ) )
2320, 22mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  <  (
# `  W )
)
24 elfzo0 11861 . . . . . . . . . . . . 13  |-  ( 1  e.  ( 0..^ (
# `  W )
)  <->  ( 1  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  1  <  ( # `  W
) ) )
2511, 17, 23, 24syl3anbrc 1180 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  e.  ( 0..^ ( # `  W
) ) )
2625ex 434 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  1  e.  ( 0..^ ( # `  W
) ) ) )
279, 26syl 16 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  (
( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  ->  1  e.  ( 0..^ ( # `  W
) ) ) )
2827adantr 465 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  (
( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  ->  1  e.  ( 0..^ ( # `  W
) ) ) )
2928impcom 430 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  1  e.  ( 0..^ ( # `  W ) ) )
30 simprr 757 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  ( W cyclShift  1 )  =  W )
31 lbfzo0 11860 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 0..^ (
# `  W )
)  <->  ( # `  W
)  e.  NN )
3231biimpri 206 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  e.  NN  ->  0  e.  ( 0..^ ( # `  W
) ) )
3313, 32sylbir 213 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
0  e.  ( 0..^ ( # `  W
) ) )
3433ex 434 . . . . . . . . . . . . . 14  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  =/=  0  ->  0  e.  ( 0..^ ( # `  W
) ) ) )
3512, 34syl5bir 218 . . . . . . . . . . . . 13  |-  ( (
# `  W )  e.  NN0  ->  ( -.  ( # `  W )  =  0  ->  0  e.  ( 0..^ ( # `  W ) ) ) )
369, 35syl 16 . . . . . . . . . . . 12  |-  ( W  e. Word  V  ->  ( -.  ( # `  W
)  =  0  -> 
0  e.  ( 0..^ ( # `  W
) ) ) )
3736adantr 465 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  ( -.  ( # `  W
)  =  0  -> 
0  e.  ( 0..^ ( # `  W
) ) ) )
3837com12 31 . . . . . . . . . 10  |-  ( -.  ( # `  W
)  =  0  -> 
( ( W  e. Word  V  /\  ( W cyclShift  1
)  =  W )  ->  0  e.  ( 0..^ ( # `  W
) ) ) )
3938adantr 465 . . . . . . . . 9  |-  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  0  e.  ( 0..^ ( # `  W
) ) ) )
4039imp 429 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  0  e.  ( 0..^ ( # `  W ) ) )
41 elfzoelz 11825 . . . . . . . . . 10  |-  ( 1  e.  ( 0..^ (
# `  W )
)  ->  1  e.  ZZ )
42 cshweqrep 12800 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  1  e.  ZZ )  ->  ( ( ( W cyclShift  1 )  =  W  /\  0  e.  ( 0..^ ( # `  W
) ) )  ->  A. i  e.  NN0  ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) ) ) )
4341, 42sylan2 474 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  1  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( ( W cyclShift  1 )  =  W  /\  0  e.  ( 0..^ ( # `  W
) ) )  ->  A. i  e.  NN0  ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) ) ) )
4443imp 429 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  1  e.  (
0..^ ( # `  W
) ) )  /\  ( ( W cyclShift  1
)  =  W  /\  0  e.  ( 0..^ ( # `  W
) ) ) )  ->  A. i  e.  NN0  ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) ) )
458, 29, 30, 40, 44syl22anc 1229 . . . . . . 7  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  A. i  e.  NN0  ( W ` 
0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `  W
) ) ) )
46 0nn0 10831 . . . . . . . . 9  |-  0  e.  NN0
47 fzossnn0 11854 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0..^ ( # `  W
) )  C_  NN0 )
48 ssralv 3560 . . . . . . . . 9  |-  ( ( 0..^ ( # `  W
) )  C_  NN0  ->  ( A. i  e.  NN0  ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) ) ) )
4946, 47, 48mp2b 10 . . . . . . . 8  |-  ( A. i  e.  NN0  ( W `
 0 )  =  ( W `  (
( 0  +  ( i  x.  1 ) )  mod  ( # `  W ) ) )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 0 )  =  ( W `  (
( 0  +  ( i  x.  1 ) )  mod  ( # `  W ) ) ) )
50 eqcom 2466 . . . . . . . . . 10  |-  ( ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  <->  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  =  ( W `
 0 ) )
51 elfzoelz 11825 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  i  e.  ZZ )
52 zre 10889 . . . . . . . . . . . . . . . . . . 19  |-  ( i  e.  ZZ  ->  i  e.  RR )
53 ax-1rid 9579 . . . . . . . . . . . . . . . . . . 19  |-  ( i  e.  RR  ->  (
i  x.  1 )  =  i )
5452, 53syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( i  e.  ZZ  ->  (
i  x.  1 )  =  i )
5554oveq2d 6312 . . . . . . . . . . . . . . . . 17  |-  ( i  e.  ZZ  ->  (
0  +  ( i  x.  1 ) )  =  ( 0  +  i ) )
56 zcn 10890 . . . . . . . . . . . . . . . . . 18  |-  ( i  e.  ZZ  ->  i  e.  CC )
5756addid2d 9798 . . . . . . . . . . . . . . . . 17  |-  ( i  e.  ZZ  ->  (
0  +  i )  =  i )
5855, 57eqtrd 2498 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ZZ  ->  (
0  +  ( i  x.  1 ) )  =  i )
5951, 58syl 16 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( 0  +  ( i  x.  1 ) )  =  i )
6059oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) )  =  ( i  mod  ( # `  W ) ) )
61 zmodidfzoimp 12028 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( i  mod  ( # `  W
) )  =  i )
6260, 61eqtrd 2498 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) )  =  i )
6362fveq2d 5876 . . . . . . . . . . . 12  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  =  ( W `
 i ) )
6463eqeq1d 2459 . . . . . . . . . . 11  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( ( W `  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) ) )  =  ( W `  0
)  <->  ( W `  i )  =  ( W `  0 ) ) )
6564biimpd 207 . . . . . . . . . 10  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( ( W `  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) ) )  =  ( W `  0
)  ->  ( W `  i )  =  ( W `  0 ) ) )
6650, 65syl5bi 217 . . . . . . . . 9  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  ->  ( W `  i )  =  ( W `  0 ) ) )
6766ralimia 2848 . . . . . . . 8  |-  ( A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 0 )  =  ( W `  (
( 0  +  ( i  x.  1 ) )  mod  ( # `  W ) ) )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
6849, 67syl 16 . . . . . . 7  |-  ( A. i  e.  NN0  ( W `
 0 )  =  ( W `  (
( 0  +  ( i  x.  1 ) )  mod  ( # `  W ) ) )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
6945, 68syl 16 . . . . . 6  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
7069ex 434 . . . . 5  |-  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
7170impancom 440 . . . 4  |-  ( ( -.  ( # `  W
)  =  0  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  ( -.  ( # `
 W )  =  1  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
72 eqid 2457 . . . . . 6  |-  ( W `
 0 )  =  ( W `  0
)
73 c0ex 9607 . . . . . . 7  |-  0  e.  _V
74 fveq2 5872 . . . . . . . 8  |-  ( i  =  0  ->  ( W `  i )  =  ( W ` 
0 ) )
7574eqeq1d 2459 . . . . . . 7  |-  ( i  =  0  ->  (
( W `  i
)  =  ( W `
 0 )  <->  ( W `  0 )  =  ( W `  0
) ) )
7673, 75ralsn 4071 . . . . . 6  |-  ( A. i  e.  { 0 }  ( W `  i )  =  ( W `  0 )  <-> 
( W `  0
)  =  ( W `
 0 ) )
7772, 76mpbir 209 . . . . 5  |-  A. i  e.  { 0 }  ( W `  i )  =  ( W ` 
0 )
78 oveq2 6304 . . . . . . 7  |-  ( (
# `  W )  =  1  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 1 ) )
79 fzo01 11899 . . . . . . 7  |-  ( 0..^ 1 )  =  {
0 }
8078, 79syl6eq 2514 . . . . . 6  |-  ( (
# `  W )  =  1  ->  (
0..^ ( # `  W
) )  =  {
0 } )
8180raleqdv 3060 . . . . 5  |-  ( (
# `  W )  =  1  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  A. i  e.  {
0 }  ( W `
 i )  =  ( W `  0
) ) )
8277, 81mpbiri 233 . . . 4  |-  ( (
# `  W )  =  1  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
8371, 82pm2.61d2 160 . . 3  |-  ( ( -.  ( # `  W
)  =  0  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
8483ex 434 . 2  |-  ( -.  ( # `  W
)  =  0  -> 
( ( W  e. Word  V  /\  ( W cyclShift  1
)  =  W )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
857, 84pm2.61i 164 1  |-  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645   NNcn 10556   NN0cn0 10816   ZZcz 10885  ..^cfzo 11820    mod cmo 11998   #chash 12407  Word cword 12537   cyclShift ccsh 12770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-hash 12408  df-word 12545  df-concat 12547  df-substr 12549  df-csh 12771
This theorem is referenced by:  cshw1repsw  12802
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