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Theorem erclwwlksym0 30475
Description: Lemma for erclwwlktr 30482 and erclwwlkntr 30498. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
Hypotheses
Ref Expression
erclwwlksym0.1  |-  ( ph  ->  y  e. Word  V )
erclwwlksym0.2  |-  ( ph  ->  ( # `  x
)  =  ( # `  y ) )
Assertion
Ref Expression
erclwwlksym0  |-  ( 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 erclwwlksym0
StepHypRef Expression
1 erclwwlksym0.2 . . . . . . 7  |-  ( ph  ->  ( # `  x
)  =  ( # `  y ) )
2 fznn0sub2 11486 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
( # `  y )  -  m )  e.  ( 0 ... ( # `
 y ) ) )
3 oveq2 6097 . . . . . . . . 9  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( 0 ... ( # `  x
) )  =  ( 0 ... ( # `  y ) ) )
43eleq2d 2508 . . . . . . . 8  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( (
( # `  y )  -  m )  e.  ( 0 ... ( # `
 x ) )  <-> 
( ( # `  y
)  -  m )  e.  ( 0 ... ( # `  y
) ) ) )
52, 4syl5ibr 221 . . . . . . 7  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( m  e.  ( 0 ... ( # `
 y ) )  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) ) )
61, 5syl 16 . . . . . 6  |-  ( ph  ->  ( m  e.  ( 0 ... ( # `  y ) )  -> 
( ( # `  y
)  -  m )  e.  ( 0 ... ( # `  x
) ) ) )
76com12 31 . . . . 5  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  ( ph  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) ) )
87adantr 465 . . . 4  |-  ( ( m  e.  ( 0 ... ( # `  y
) )  /\  x  =  ( y cyclShift  m
) )  ->  ( ph  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) ) )
98impcom 430 . . 3  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  ( ( # `  y )  -  m
)  e.  ( 0 ... ( # `  x
) ) )
10 erclwwlksym0.1 . . . . . . . 8  |-  ( ph  ->  y  e. Word  V )
11 simpl 457 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  -> 
y  e. Word  V )
12 elfzelz 11451 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  m  e.  ZZ )
1312adantl 466 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  ->  m  e.  ZZ )
14 elfz2nn0 11478 . . . . . . . . . . 11  |-  ( m  e.  ( 0 ... ( # `  y
) )  <->  ( m  e.  NN0  /\  ( # `  y )  e.  NN0  /\  m  <_  ( # `  y
) ) )
15 nn0z 10667 . . . . . . . . . . . . 13  |-  ( (
# `  y )  e.  NN0  ->  ( # `  y
)  e.  ZZ )
16 nn0z 10667 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  ->  m  e.  ZZ )
17 zsubcl 10685 . . . . . . . . . . . . 13  |-  ( ( ( # `  y
)  e.  ZZ  /\  m  e.  ZZ )  ->  ( ( # `  y
)  -  m )  e.  ZZ )
1815, 16, 17syl2anr 478 . . . . . . . . . . . 12  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0 )  -> 
( ( # `  y
)  -  m )  e.  ZZ )
19183adant3 1008 . . . . . . . . . . 11  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0  /\  m  <_  ( # `  y
) )  ->  (
( # `  y )  -  m )  e.  ZZ )
2014, 19sylbi 195 . . . . . . . . . 10  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
( # `  y )  -  m )  e.  ZZ )
2120adantl 466 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  -> 
( ( # `  y
)  -  m )  e.  ZZ )
2211, 13, 213jca 1168 . . . . . . . 8  |-  ( ( y  e. Word  V  /\  m  e.  ( 0 ... ( # `  y
) ) )  -> 
( y  e. Word  V  /\  m  e.  ZZ  /\  ( ( # `  y
)  -  m )  e.  ZZ ) )
2310, 22sylan 471 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( y  e. Word  V  /\  m  e.  ZZ  /\  ( (
# `  y )  -  m )  e.  ZZ ) )
24 2cshw 12445 . . . . . . 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 16 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( (
y cyclShift  m ) cyclShift  ( ( # `
 y )  -  m ) )  =  ( y cyclShift  ( m  +  ( ( # `  y )  -  m
) ) ) )
26 nn0cn 10587 . . . . . . . . . . . 12  |-  ( m  e.  NN0  ->  m  e.  CC )
27 nn0cn 10587 . . . . . . . . . . . 12  |-  ( (
# `  y )  e.  NN0  ->  ( # `  y
)  e.  CC )
2826, 27anim12i 566 . . . . . . . . . . 11  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0 )  -> 
( m  e.  CC  /\  ( # `  y
)  e.  CC ) )
29283adant3 1008 . . . . . . . . . 10  |-  ( ( m  e.  NN0  /\  ( # `  y )  e.  NN0  /\  m  <_  ( # `  y
) )  ->  (
m  e.  CC  /\  ( # `  y )  e.  CC ) )
3014, 29sylbi 195 . . . . . . . . 9  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
m  e.  CC  /\  ( # `  y )  e.  CC ) )
31 pncan3 9616 . . . . . . . . 9  |-  ( ( m  e.  CC  /\  ( # `  y )  e.  CC )  -> 
( m  +  ( ( # `  y
)  -  m ) )  =  ( # `  y ) )
3230, 31syl 16 . . . . . . . 8  |-  ( m  e.  ( 0 ... ( # `  y
) )  ->  (
m  +  ( (
# `  y )  -  m ) )  =  ( # `  y
) )
3332adantl 466 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( m  +  ( ( # `  y )  -  m
) )  =  (
# `  y )
)
3433oveq2d 6105 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( y cyclShift  ( m  +  ( (
# `  y )  -  m ) ) )  =  ( y cyclShift  ( # `
 y ) ) )
35 cshwn 12432 . . . . . . . 8  |-  ( y  e. Word  V  ->  (
y cyclShift  ( # `  y
) )  =  y )
3610, 35syl 16 . . . . . . 7  |-  ( ph  ->  ( y cyclShift  ( # `  y
) )  =  y )
3736adantr 465 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  ( y cyclShift  (
# `  y )
)  =  y )
3825, 34, 373eqtrrd 2478 . . . . 5  |-  ( (
ph  /\  m  e.  ( 0 ... ( # `
 y ) ) )  ->  y  =  ( ( y cyclShift  m
) cyclShift  ( ( # `  y
)  -  m ) ) )
3938adantrr 716 . . . 4  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  y  =  ( ( y cyclShift  m ) cyclShift  ( ( # `  y
)  -  m ) ) )
40 oveq1 6096 . . . . . . 7  |-  ( x  =  ( y cyclShift  m
)  ->  ( x cyclShift  ( ( # `  y
)  -  m ) )  =  ( ( y cyclShift  m ) cyclShift  ( ( # `
 y )  -  m ) ) )
4140eqeq2d 2452 . . . . . 6  |-  ( x  =  ( y cyclShift  m
)  ->  ( y  =  ( x cyclShift  (
( # `  y )  -  m ) )  <-> 
y  =  ( ( y cyclShift  m ) cyclShift  ( ( # `
 y )  -  m ) ) ) )
4241adantl 466 . . . . 5  |-  ( ( m  e.  ( 0 ... ( # `  y
) )  /\  x  =  ( y cyclShift  m
) )  ->  (
y  =  ( x cyclShift  ( ( # `  y
)  -  m ) )  <->  y  =  ( ( y cyclShift  m ) cyclShift  ( ( # `  y
)  -  m ) ) ) )
4342adantl 466 . . . 4  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  ( y  =  ( x cyclShift  ( ( # `
 y )  -  m ) )  <->  y  =  ( ( y cyclShift  m
) cyclShift  ( ( # `  y
)  -  m ) ) ) )
4439, 43mpbird 232 . . 3  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  y  =  ( x cyclShift  ( ( # `  y
)  -  m ) ) )
45 oveq2 6097 . . . . 5  |-  ( n  =  ( ( # `  y )  -  m
)  ->  ( x cyclShift  n )  =  ( x cyclShift  ( ( # `  y
)  -  m ) ) )
4645eqeq2d 2452 . . . 4  |-  ( n  =  ( ( # `  y )  -  m
)  ->  ( y  =  ( x cyclShift  n
)  <->  y  =  ( x cyclShift  ( ( # `  y
)  -  m ) ) ) )
4746rspcev 3071 . . 3  |-  ( ( ( ( # `  y
)  -  m )  e.  ( 0 ... ( # `  x
) )  /\  y  =  ( x cyclShift  (
( # `  y )  -  m ) ) )  ->  E. n  e.  ( 0 ... ( # `
 x ) ) y  =  ( x cyclShift  n ) )
489, 44, 47syl2anc 661 . 2  |-  ( (
ph  /\  ( m  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  m ) ) )  ->  E. n  e.  ( 0 ... ( # `  x ) ) y  =  ( x cyclShift  n
) )
4948ex 434 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2714   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280    + caddc 9283    <_ cle 9417    - cmin 9593   NN0cn0 10577   ZZcz 10644   ...cfz 11435   #chash 12101  Word cword 12219   cyclShift ccsh 12423
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-hash 12102  df-word 12227  df-concat 12229  df-substr 12231  df-csh 12424
This theorem is referenced by:  erclwwlksym  30481  erclwwlknsym  30497
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