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Theorem cshw1 12456
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 3784 . . . 4  |-  A. i  e.  (/)  ( W `  i )  =  ( W `  0 )
2 oveq2 6099 . . . . . 6  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 0 ) )
3 fzo0 11573 . . . . . 6  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2491 . . . . 5  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  (/) )
54raleqdv 2923 . . . 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 755 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  W  e. Word  V )
9 lencl 12249 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
10 1nn0 10595 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1110a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  e.  NN0 )
12 df-ne 2608 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  0  <->  -.  ( # `  W
)  =  0 )
13 elnnne0 10593 . . . . . . . . . . . . . . . . 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 2608 . . . . . . . . . . . . . . . 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 10349 . . . . . . . . . . . . . . 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 11587 . . . . . . . . . . . . 13  |-  ( 1  e.  ( 0..^ (
# `  W )
)  <->  ( 1  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  1  <  ( # `  W
) ) )
2511, 17, 23, 24syl3anbrc 1172 . . . . . . . . . . . 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 756 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  ( W cyclShift  1 )  =  W )
31 lbfzo0 11586 . . . . . . . . . . . . . . . . 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 11553 . . . . . . . . . 10  |-  ( 1  e.  ( 0..^ (
# `  W )
)  ->  1  e.  ZZ )
42 cshweqrep 12455 . . . . . . . . . 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 1219 . . . . . . 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 10594 . . . . . . . . 9  |-  0  e.  NN0
47 fzossnn0 11580 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0..^ ( # `  W
) )  C_  NN0 )
48 ssralv 3416 . . . . . . . . 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 2445 . . . . . . . . . 10  |-  ( ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  <->  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  =  ( W `
 0 ) )
51 elfzoelz 11553 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  i  e.  ZZ )
52 zre 10650 . . . . . . . . . . . . . . . . . . 19  |-  ( i  e.  ZZ  ->  i  e.  RR )
53 ax-1rid 9352 . . . . . . . . . . . . . . . . . . 19  |-  ( i  e.  RR  ->  (
i  x.  1 )  =  i )
5452, 53syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( i  e.  ZZ  ->  (
i  x.  1 )  =  i )
5554oveq2d 6107 . . . . . . . . . . . . . . . . 17  |-  ( i  e.  ZZ  ->  (
0  +  ( i  x.  1 ) )  =  ( 0  +  i ) )
56 zcn 10651 . . . . . . . . . . . . . . . . . 18  |-  ( i  e.  ZZ  ->  i  e.  CC )
5756addid2d 9570 . . . . . . . . . . . . . . . . 17  |-  ( i  e.  ZZ  ->  (
0  +  i )  =  i )
5855, 57eqtrd 2475 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ZZ  ->  (
0  +  ( i  x.  1 ) )  =  i )
5951, 58syl 16 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( 0  +  ( i  x.  1 ) )  =  i )
6059oveq1d 6106 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) )  =  ( i  mod  ( # `  W ) ) )
61 zmodidfzoimp 11738 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( i  mod  ( # `  W
) )  =  i )
6260, 61eqtrd 2475 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) )  =  i )
6362fveq2d 5695 . . . . . . . . . . . 12  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  =  ( W `
 i ) )
6463eqeq1d 2451 . . . . . . . . . . 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 2789 . . . . . . . 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 2443 . . . . . 6  |-  ( W `
 0 )  =  ( W `  0
)
73 c0ex 9380 . . . . . . 7  |-  0  e.  _V
74 fveq2 5691 . . . . . . . 8  |-  ( i  =  0  ->  ( W `  i )  =  ( W ` 
0 ) )
7574eqeq1d 2451 . . . . . . 7  |-  ( i  =  0  ->  (
( W `  i
)  =  ( W `
 0 )  <->  ( W `  0 )  =  ( W `  0
) ) )
7673, 75ralsn 3915 . . . . . 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 6099 . . . . . . 7  |-  ( (
# `  W )  =  1  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 1 ) )
79 fzo01 11612 . . . . . . 7  |-  ( 0..^ 1 )  =  {
0 }
8078, 79syl6eq 2491 . . . . . 6  |-  ( (
# `  W )  =  1  ->  (
0..^ ( # `  W
) )  =  {
0 } )
8180raleqdv 2923 . . . . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    C_ wss 3328   (/)c0 3637   {csn 3877   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418   NNcn 10322   NN0cn0 10579   ZZcz 10646  ..^cfzo 11548    mod cmo 11708   #chash 12103  Word cword 12221   cyclShift ccsh 12425
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-hash 12104  df-word 12229  df-concat 12231  df-substr 12233  df-csh 12426
This theorem is referenced by:  cshw1repsw  12457
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