MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cshw1 Structured version   Unicode version

Theorem cshw1 12911
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 3902 . . . 4  |-  A. i  e.  (/)  ( W `  i )  =  ( W `  0 )
2 oveq2 6309 . . . . . 6  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 0 ) )
3 fzo0 11942 . . . . . 6  |-  ( 0..^ 0 )  =  (/)
42, 3syl6eq 2479 . . . . 5  |-  ( (
# `  W )  =  0  ->  (
0..^ ( # `  W
) )  =  (/) )
54raleqdv 3031 . . . 4  |-  ( (
# `  W )  =  0  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  A. i  e.  (/)  ( W `  i )  =  ( W ` 
0 ) ) )
61, 5mpbiri 236 . . 3  |-  ( (
# `  W )  =  0  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
76a1d 26 . 2  |-  ( (
# `  W )  =  0  ->  (
( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
8 simprl 762 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  W  e. Word  V )
9 lencl 12679 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
10 1nn0 10885 . . . . . . . . . . . . . 14  |-  1  e.  NN0
1110a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  e.  NN0 )
12 df-ne 2620 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  0  <->  -.  ( # `  W
)  =  0 )
13 elnnne0 10883 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  NN0  /\  ( # `  W
)  =/=  0 ) )
1413simplbi2com 631 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  0  ->  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  NN ) )
1512, 14sylbir 216 . . . . . . . . . . . . . . 15  |-  ( -.  ( # `  W
)  =  0  -> 
( ( # `  W
)  e.  NN0  ->  (
# `  W )  e.  NN ) )
1615adantr 466 . . . . . . . . . . . . . 14  |-  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  ( ( # `  W )  e.  NN0  ->  ( # `  W
)  e.  NN ) )
1716impcom 431 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  ( # `  W
)  e.  NN )
18 df-ne 2620 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  1  <->  -.  ( # `  W
)  =  1 )
1918biimpri 209 . . . . . . . . . . . . . . 15  |-  ( -.  ( # `  W
)  =  1  -> 
( # `  W )  =/=  1 )
2019ad2antll 733 . . . . . . . . . . . . . 14  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  ( # `  W
)  =/=  1 )
21 nngt1ne1 10636 . . . . . . . . . . . . . . 15  |-  ( (
# `  W )  e.  NN  ->  ( 1  <  ( # `  W
)  <->  ( # `  W
)  =/=  1 ) )
2217, 21syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  ( 1  <  ( # `  W
)  <->  ( # `  W
)  =/=  1 ) )
2320, 22mpbird 235 . . . . . . . . . . . . 13  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  <  (
# `  W )
)
24 elfzo0 11956 . . . . . . . . . . . . 13  |-  ( 1  e.  ( 0..^ (
# `  W )
)  <->  ( 1  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  1  <  ( # `  W
) ) )
2511, 17, 23, 24syl3anbrc 1189 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  e.  NN0  /\  ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 ) )  ->  1  e.  ( 0..^ ( # `  W
) ) )
2625ex 435 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  1  e.  ( 0..^ ( # `  W
) ) ) )
279, 26syl 17 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  (
( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  ->  1  e.  ( 0..^ ( # `  W
) ) ) )
2827adantr 466 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  (
( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  ->  1  e.  ( 0..^ ( # `  W
) ) ) )
2928impcom 431 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  1  e.  ( 0..^ ( # `  W ) ) )
30 simprr 764 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  ( W cyclShift  1 )  =  W )
31 lbfzo0 11955 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 0..^ (
# `  W )
)  <->  ( # `  W
)  e.  NN )
3231biimpri 209 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  e.  NN  ->  0  e.  ( 0..^ ( # `  W
) ) )
3313, 32sylbir 216 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
0  e.  ( 0..^ ( # `  W
) ) )
3433ex 435 . . . . . . . . . . . . . 14  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  =/=  0  ->  0  e.  ( 0..^ ( # `  W
) ) ) )
3512, 34syl5bir 221 . . . . . . . . . . . . 13  |-  ( (
# `  W )  e.  NN0  ->  ( -.  ( # `  W )  =  0  ->  0  e.  ( 0..^ ( # `  W ) ) ) )
369, 35syl 17 . . . . . . . . . . . 12  |-  ( W  e. Word  V  ->  ( -.  ( # `  W
)  =  0  -> 
0  e.  ( 0..^ ( # `  W
) ) ) )
3736adantr 466 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  ( -.  ( # `  W
)  =  0  -> 
0  e.  ( 0..^ ( # `  W
) ) ) )
3837com12 32 . . . . . . . . . 10  |-  ( -.  ( # `  W
)  =  0  -> 
( ( W  e. Word  V  /\  ( W cyclShift  1
)  =  W )  ->  0  e.  ( 0..^ ( # `  W
) ) ) )
3938adantr 466 . . . . . . . . 9  |-  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  0  e.  ( 0..^ ( # `  W
) ) ) )
4039imp 430 . . . . . . . 8  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  0  e.  ( 0..^ ( # `  W ) ) )
41 elfzoelz 11920 . . . . . . . . . 10  |-  ( 1  e.  ( 0..^ (
# `  W )
)  ->  1  e.  ZZ )
42 cshweqrep 12910 . . . . . . . . . 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 476 . . . . . . . . 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 430 . . . . . . . 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 1265 . . . . . . 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 10884 . . . . . . . . 9  |-  0  e.  NN0
47 fzossnn0 11949 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( 0..^ ( # `  W
) )  C_  NN0 )
48 ssralv 3525 . . . . . . . . 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 2431 . . . . . . . . . 10  |-  ( ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  <->  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  =  ( W `
 0 ) )
51 elfzoelz 11920 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  i  e.  ZZ )
52 zre 10941 . . . . . . . . . . . . . . . . . . 19  |-  ( i  e.  ZZ  ->  i  e.  RR )
53 ax-1rid 9609 . . . . . . . . . . . . . . . . . . 19  |-  ( i  e.  RR  ->  (
i  x.  1 )  =  i )
5452, 53syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( i  e.  ZZ  ->  (
i  x.  1 )  =  i )
5554oveq2d 6317 . . . . . . . . . . . . . . . . 17  |-  ( i  e.  ZZ  ->  (
0  +  ( i  x.  1 ) )  =  ( 0  +  i ) )
56 zcn 10942 . . . . . . . . . . . . . . . . . 18  |-  ( i  e.  ZZ  ->  i  e.  CC )
5756addid2d 9834 . . . . . . . . . . . . . . . . 17  |-  ( i  e.  ZZ  ->  (
0  +  i )  =  i )
5855, 57eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ZZ  ->  (
0  +  ( i  x.  1 ) )  =  i )
5951, 58syl 17 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( 0  +  ( i  x.  1 ) )  =  i )
6059oveq1d 6316 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) )  =  ( i  mod  ( # `  W ) ) )
61 zmodidfzoimp 12126 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( i  mod  ( # `  W
) )  =  i )
6260, 61eqtrd 2463 . . . . . . . . . . . . 13  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) )  =  i )
6362fveq2d 5881 . . . . . . . . . . . 12  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  =  ( W `
 i ) )
6463eqeq1d 2424 . . . . . . . . . . 11  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( ( W `  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) ) )  =  ( W `  0
)  <->  ( W `  i )  =  ( W `  0 ) ) )
6564biimpd 210 . . . . . . . . . 10  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( ( W `  ( (
0  +  ( i  x.  1 ) )  mod  ( # `  W
) ) )  =  ( W `  0
)  ->  ( W `  i )  =  ( W `  0 ) ) )
6650, 65syl5bi 220 . . . . . . . . 9  |-  ( i  e.  ( 0..^ (
# `  W )
)  ->  ( ( W `  0 )  =  ( W `  ( ( 0  +  ( i  x.  1 ) )  mod  ( # `
 W ) ) )  ->  ( W `  i )  =  ( W `  0 ) ) )
6766ralimia 2816 . . . . . . . 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 17 . . . . . . 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 17 . . . . . 6  |-  ( ( ( -.  ( # `  W )  =  0  /\  -.  ( # `  W )  =  1 )  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
7069ex 435 . . . . 5  |-  ( ( -.  ( # `  W
)  =  0  /\ 
-.  ( # `  W
)  =  1 )  ->  ( ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
7170impancom 441 . . . 4  |-  ( ( -.  ( # `  W
)  =  0  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  ( -.  ( # `
 W )  =  1  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
72 eqid 2422 . . . . . 6  |-  ( W `
 0 )  =  ( W `  0
)
73 c0ex 9637 . . . . . . 7  |-  0  e.  _V
74 fveq2 5877 . . . . . . . 8  |-  ( i  =  0  ->  ( W `  i )  =  ( W ` 
0 ) )
7574eqeq1d 2424 . . . . . . 7  |-  ( i  =  0  ->  (
( W `  i
)  =  ( W `
 0 )  <->  ( W `  0 )  =  ( W `  0
) ) )
7673, 75ralsn 4035 . . . . . 6  |-  ( A. i  e.  { 0 }  ( W `  i )  =  ( W `  0 )  <-> 
( W `  0
)  =  ( W `
 0 ) )
7772, 76mpbir 212 . . . . 5  |-  A. i  e.  { 0 }  ( W `  i )  =  ( W ` 
0 )
78 oveq2 6309 . . . . . . 7  |-  ( (
# `  W )  =  1  ->  (
0..^ ( # `  W
) )  =  ( 0..^ 1 ) )
79 fzo01 11994 . . . . . . 7  |-  ( 0..^ 1 )  =  {
0 }
8078, 79syl6eq 2479 . . . . . 6  |-  ( (
# `  W )  =  1  ->  (
0..^ ( # `  W
) )  =  {
0 } )
8180raleqdv 3031 . . . . 5  |-  ( (
# `  W )  =  1  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  <->  A. i  e.  {
0 }  ( W `
 i )  =  ( W `  0
) ) )
8277, 81mpbiri 236 . . . 4  |-  ( (
# `  W )  =  1  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
8371, 82pm2.61d2 163 . . 3  |-  ( ( -.  ( # `  W
)  =  0  /\  ( W  e. Word  V  /\  ( W cyclShift  1 )  =  W ) )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
8483ex 435 . 2  |-  ( -.  ( # `  W
)  =  0  -> 
( ( W  e. Word  V  /\  ( W cyclShift  1
)  =  W )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
857, 84pm2.61i 167 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 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775    C_ wss 3436   (/)c0 3761   {csn 3996   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675   NNcn 10609   NN0cn0 10869   ZZcz 10937  ..^cfzo 11915    mod cmo 12095   #chash 12514  Word cword 12648   cyclShift ccsh 12880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-hash 12515  df-word 12656  df-concat 12658  df-substr 12660  df-csh 12881
This theorem is referenced by:  cshw1repsw  12912
  Copyright terms: Public domain W3C validator