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Theorem cshwcshid 12911
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlktr 25388 and erclwwlkntr 25400. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
Hypotheses
Ref Expression
cshwcshid.1  |-  ( ph  ->  y  e. Word  V )
cshwcshid.2  |-  ( ph  ->  ( # `  x
)  =  ( # `  y ) )
Assertion
Ref Expression
cshwcshid  |-  ( ph  ->  ( ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) )  ->  E. n  e.  (
0 ... ( # `  x
) ) y  =  ( x cyclShift  n )
) )
Distinct variable group:    m, n, x, y
Allowed substitution hints:    ph( x, y, m, n)    V( x, y, m, n)

Proof of Theorem cshwcshid
StepHypRef Expression
1 cshwcshid.2 . . . . . . 7  |-  ( ph  ->  ( # `  x
)  =  ( # `  y ) )
2 fznn0sub2 11895 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
( # `  y )  -  m )  e.  ( 0 ... ( # `
 y ) ) )
3 oveq2 6313 . . . . . . . . 9  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( 0 ... ( # `  x
) )  =  ( 0 ... ( # `  y ) ) )
43eleq2d 2499 . . . . . . . 8  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( (
( # `  y )  -  m )  e.  ( 0 ... ( # `
 x ) )  <-> 
( ( # `  y
)  -  m )  e.  ( 0 ... ( # `  y
) ) ) )
52, 4syl5ibr 224 . . . . . . 7  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( m  e.  ( 0 ... ( # `
 y ) )  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) ) )
61, 5syl 17 . . . . . 6  |-  ( ph  ->  ( m  e.  ( 0 ... ( # `  y ) )  -> 
( ( # `  y
)  -  m )  e.  ( 0 ... ( # `  x
) ) ) )
76com12 32 . . . . 5  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  ( ph  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) ) )
87adantr 466 . . . 4  |-  ( ( m  e.  ( 0 ... ( # `  y
) )  /\  x  =  ( y cyclShift  m
) )  ->  ( ph  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) ) )
98impcom 431 . . 3  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) )
10 cshwcshid.1 . . . . . . . 8  |-  ( ph  ->  y  e. Word  V )
11 simpl 458 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  -> 
y  e. Word  V )
12 elfzelz 11798 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  m  e.  ZZ )
1312adantl 467 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  ->  m  e.  ZZ )
14 elfz2nn0 11883 . . . . . . . . . . 11  |-  ( m  e.  ( 0 ... ( # `  y
) )  <->  ( m  e.  NN0  /\  ( # `  y )  e.  NN0  /\  m  <_  ( # `  y
) ) )
15 nn0z 10960 . . . . . . . . . . . . 13  |-  ( (
# `  y )  e.  NN0  ->  ( # `  y
)  e.  ZZ )
16 nn0z 10960 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  ->  m  e.  ZZ )
17 zsubcl 10979 . . . . . . . . . . . . 13  |-  ( ( ( # `  y
)  e.  ZZ  /\  m  e.  ZZ )  ->  ( ( # `  y
)  -  m )  e.  ZZ )
1815, 16, 17syl2anr 480 . . . . . . . . . . . 12  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0 )  -> 
( ( # `  y
)  -  m )  e.  ZZ )
19183adant3 1025 . . . . . . . . . . 11  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0  /\  m  <_  ( # `  y
) )  ->  (
( # `  y )  -  m )  e.  ZZ )
2014, 19sylbi 198 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
( # `  y )  -  m )  e.  ZZ )
2120adantl 467 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  -> 
( ( # `  y
)  -  m )  e.  ZZ )
2211, 13, 213jca 1185 . . . . . . . 8  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  -> 
( y  e. Word  V  /\  m  e.  ZZ  /\  ( ( # `  y
)  -  m )  e.  ZZ ) )
2310, 22sylan 473 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( y  e. Word  V  /\  m  e.  ZZ  /\  ( (
# `  y )  -  m )  e.  ZZ ) )
24 2cshw 12897 . . . . . . 7  |-  ( ( y  e. Word  V  /\  m  e.  ZZ  /\  (
( # `  y )  -  m )  e.  ZZ )  ->  (
( y cyclShift  m ) cyclShift  ( ( # `  y
)  -  m ) )  =  ( y cyclShift  ( m  +  (
( # `  y )  -  m ) ) ) )
2523, 24syl 17 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( (
y cyclShift  m ) cyclShift  ( ( # `
 y )  -  m ) )  =  ( y cyclShift  ( m  +  ( ( # `  y )  -  m
) ) ) )
26 nn0cn 10879 . . . . . . . . . . . 12  |-  ( m  e.  NN0  ->  m  e.  CC )
27 nn0cn 10879 . . . . . . . . . . . 12  |-  ( (
# `  y )  e.  NN0  ->  ( # `  y
)  e.  CC )
2826, 27anim12i 568 . . . . . . . . . . 11  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0 )  -> 
( m  e.  CC  /\  ( # `  y
)  e.  CC ) )
29283adant3 1025 . . . . . . . . . 10  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0  /\  m  <_  ( # `  y
) )  ->  (
m  e.  CC  /\  ( # `  y )  e.  CC ) )
3014, 29sylbi 198 . . . . . . . . 9  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
m  e.  CC  /\  ( # `  y )  e.  CC ) )
31 pncan3 9882 . . . . . . . . 9  |-  ( ( m  e.  CC  /\  ( # `  y )  e.  CC )  -> 
( m  +  ( ( # `  y
)  -  m ) )  =  ( # `  y ) )
3230, 31syl 17 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
m  +  ( (
# `  y )  -  m ) )  =  ( # `  y
) )
3332adantl 467 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( m  +  ( ( # `  y )  -  m
) )  =  (
# `  y )
)
3433oveq2d 6321 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( y cyclShift  ( m  +  ( (
# `  y )  -  m ) ) )  =  ( y cyclShift  ( # `
 y ) ) )
35 cshwn 12884 . . . . . . . 8  |-  ( y  e. Word  V  ->  (
y cyclShift  ( # `  y
) )  =  y )
3610, 35syl 17 . . . . . . 7  |-  ( ph  ->  ( y cyclShift  ( # `  y
) )  =  y )
3736adantr 466 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( y cyclShift  (
# `  y )
)  =  y )
3825, 34, 373eqtrrd 2475 . . . . 5  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  y  =  ( ( y cyclShift  m
) cyclShift  ( ( # `  y
)  -  m ) ) )
3938adantrr 721 . . . 4  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  y  =  ( ( y cyclShift  m ) cyclShift  ( ( # `  y
)  -  m ) ) )
40 oveq1 6312 . . . . . . 7  |-  ( x  =  ( y cyclShift  m
)  ->  ( x cyclShift  ( ( # `  y
)  -  m ) )  =  ( ( y cyclShift  m ) cyclShift  ( ( # `
 y )  -  m ) ) )
4140eqeq2d 2443 . . . . . 6  |-  ( x  =  ( y cyclShift  m
)  ->  ( y  =  ( x cyclShift  (
( # `  y )  -  m ) )  <-> 
y  =  ( ( y cyclShift  m ) cyclShift  ( ( # `
 y )  -  m ) ) ) )
4241adantl 467 . . . . 5  |-  ( ( m  e.  ( 0 ... ( # `  y
) )  /\  x  =  ( y cyclShift  m
) )  ->  (
y  =  ( x cyclShift  ( ( # `  y
)  -  m ) )  <->  y  =  ( ( y cyclShift  m ) cyclShift  ( ( # `  y
)  -  m ) ) ) )
4342adantl 467 . . . 4  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  ( y  =  ( x cyclShift  ( ( # `
 y )  -  m ) )  <->  y  =  ( ( y cyclShift  m
) cyclShift  ( ( # `  y
)  -  m ) ) ) )
4439, 43mpbird 235 . . 3  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  y  =  ( x cyclShift  ( ( # `  y
)  -  m ) ) )
45 oveq2 6313 . . . . 5  |-  ( n  =  ( ( # `  y )  -  m
)  ->  ( x cyclShift  n )  =  ( x cyclShift  ( ( # `  y
)  -  m ) ) )
4645eqeq2d 2443 . . . 4  |-  ( n  =  ( ( # `  y )  -  m
)  ->  ( y  =  ( x cyclShift  n
)  <->  y  =  ( x cyclShift  ( ( # `  y
)  -  m ) ) ) )
4746rspcev 3188 . . 3  |-  ( ( ( ( # `  y
)  -  m )  e.  ( 0 ... ( # `  x
) )  /\  y  =  ( x cyclShift  (
( # `  y )  -  m ) ) )  ->  E. n  e.  ( 0 ... ( # `
 x ) ) y  =  ( x cyclShift  n ) )
489, 44, 47syl2anc 665 . 2  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  E. n  e.  ( 0 ... ( # `  x ) ) y  =  ( x cyclShift  n
) )
4948ex 435 1  |-  ( ph  ->  ( ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) )  ->  E. n  e.  (
0 ... ( # `  x
) ) y  =  ( x cyclShift  n )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538    + caddc 9541    <_ cle 9675    - cmin 9859   NN0cn0 10869   ZZcz 10937   ...cfz 11782   #chash 12512  Word cword 12643   cyclShift ccsh 12875
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-hash 12513  df-word 12651  df-concat 12653  df-substr 12655  df-csh 12876
This theorem is referenced by:  erclwwlksym  25387  erclwwlknsym  25399
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