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Theorem 2cshwid 28280
Description: Cyclically shifting a word two times resulting in the word itself. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 5-Jun-2018.)
Assertion
Ref Expression
2cshwid  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( ( W CyclShift  N ) CyclShift  ( ( # `  W
)  -  N ) )  =  W )

Proof of Theorem 2cshwid
StepHypRef Expression
1 id 21 . . . 4  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  e.  ( 0 ... ( # `
 W ) ) )
2 fznn0sub2 11088 . . . 4  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  (
( # `  W )  -  N )  e.  ( 0 ... ( # `
 W ) ) )
3 elfz2nn0 11084 . . . . 5  |-  ( N  e.  ( 0 ... ( # `  W
) )  <->  ( N  e.  NN0  /\  ( # `  W )  e.  NN0  /\  N  <_  ( # `  W
) ) )
4 nn0cn 10233 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
5 nn0cn 10233 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  CC )
6 pncan3 9315 . . . . . . . 8  |-  ( ( N  e.  CC  /\  ( # `  W )  e.  CC )  -> 
( N  +  ( ( # `  W
)  -  N ) )  =  ( # `  W ) )
74, 5, 6syl2an 465 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( # `  W )  e.  NN0 )  -> 
( N  +  ( ( # `  W
)  -  N ) )  =  ( # `  W ) )
8 nn0re 10232 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
98leidd 9595 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  <_  ( # `  W
) )
109adantl 454 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( # `  W )  e.  NN0 )  -> 
( # `  W )  <_  ( # `  W
) )
117, 10eqbrtrd 4234 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( # `  W )  e.  NN0 )  -> 
( N  +  ( ( # `  W
)  -  N ) )  <_  ( # `  W
) )
12113adant3 978 . . . . 5  |-  ( ( N  e.  NN0  /\  ( # `  W )  e.  NN0  /\  N  <_ 
( # `  W ) )  ->  ( N  +  ( ( # `  W )  -  N
) )  <_  ( # `
 W ) )
133, 12sylbi 189 . . . 4  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  ( N  +  ( ( # `
 W )  -  N ) )  <_ 
( # `  W ) )
141, 2, 133jca 1135 . . 3  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  ( N  e.  ( 0 ... ( # `  W
) )  /\  (
( # `  W )  -  N )  e.  ( 0 ... ( # `
 W ) )  /\  ( N  +  ( ( # `  W
)  -  N ) )  <_  ( # `  W
) ) )
15 2cshw1 28273 . . . 4  |-  ( W  e. Word  V  ->  (
( N  e.  ( 0 ... ( # `  W ) )  /\  ( ( # `  W
)  -  N )  e.  ( 0 ... ( # `  W
) )  /\  ( N  +  ( ( # `
 W )  -  N ) )  <_ 
( # `  W ) )  ->  ( ( W CyclShift  N ) CyclShift  ( ( # `
 W )  -  N ) )  =  ( W CyclShift  ( N  +  ( ( # `  W )  -  N
) ) ) ) )
1615imp 420 . . 3  |-  ( ( W  e. Word  V  /\  ( N  e.  (
0 ... ( # `  W
) )  /\  (
( # `  W )  -  N )  e.  ( 0 ... ( # `
 W ) )  /\  ( N  +  ( ( # `  W
)  -  N ) )  <_  ( # `  W
) ) )  -> 
( ( W CyclShift  N ) CyclShift  ( ( # `  W
)  -  N ) )  =  ( W CyclShift  ( N  +  (
( # `  W )  -  N ) ) ) )
1714, 16sylan2 462 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( ( W CyclShift  N ) CyclShift  ( ( # `  W
)  -  N ) )  =  ( W CyclShift  ( N  +  (
( # `  W )  -  N ) ) ) )
1873adant3 978 . . . . 5  |-  ( ( N  e.  NN0  /\  ( # `  W )  e.  NN0  /\  N  <_ 
( # `  W ) )  ->  ( N  +  ( ( # `  W )  -  N
) )  =  (
# `  W )
)
193, 18sylbi 189 . . . 4  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  ( N  +  ( ( # `
 W )  -  N ) )  =  ( # `  W
) )
2019adantl 454 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( N  +  ( ( # `  W
)  -  N ) )  =  ( # `  W ) )
2120oveq2d 6099 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( W CyclShift  ( N  +  ( ( # `  W
)  -  N ) ) )  =  ( W CyclShift  ( # `  W
) ) )
22 cshwn 28261 . . 3  |-  ( W  e. Word  V  ->  ( W CyclShift  ( # `  W
) )  =  W )
2322adantr 453 . 2  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( W CyclShift  ( # `  W
) )  =  W )
2417, 21, 233eqtrd 2474 1  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( ( W CyclShift  N ) CyclShift  ( ( # `  W
)  -  N ) )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    + caddc 8995    <_ cle 9123    - cmin 9293   NN0cn0 10223   ...cfz 11045   #chash 11620  Word cword 11719   CyclShift ccsh 28252
This theorem is referenced by:  3cshw  28291  cshwsym  28295  cshwssizelem4a  28305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-substr 11728  df-csh 28254
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