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Theorem erclwwlknref 24616
Description:  .~ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
erclwwlkn.r  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
Assertion
Ref Expression
erclwwlknref  |-  ( x  e.  W  <->  x  .~  x )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    x, n, t, u    n, N
Allowed substitution hints:    .~ ( x, u, t, n)    E( x, n)    N( x)    V( x)    W( x, n)

Proof of Theorem erclwwlknref
StepHypRef Expression
1 df-3an 975 . . 3  |-  ( ( x  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( x cyclShift  n
) )  <->  ( (
x  e.  W  /\  x  e.  W )  /\  E. n  e.  ( 0 ... N ) x  =  ( x cyclShift  n ) ) )
2 anidm 644 . . . 4  |-  ( ( x  e.  W  /\  x  e.  W )  <->  x  e.  W )
32anbi1i 695 . . 3  |-  ( ( ( x  e.  W  /\  x  e.  W
)  /\  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) )  <->  ( x  e.  W  /\  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) ) )
41, 3bitri 249 . 2  |-  ( ( x  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( x cyclShift  n
) )  <->  ( x  e.  W  /\  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) ) )
5 vex 3121 . . 3  |-  x  e. 
_V
6 erclwwlkn.w . . . 4  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
7 erclwwlkn.r . . . 4  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
86, 7erclwwlkneq 24614 . . 3  |-  ( ( x  e.  _V  /\  x  e.  _V )  ->  ( x  .~  x  <->  ( x  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( x cyclShift  n
) ) ) )
95, 5, 8mp2an 672 . 2  |-  ( x  .~  x  <->  ( x  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) ) )
10 clwwlknprop 24563 . . . . 5  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
11 cshw0 12740 . . . . . . 7  |-  ( x  e. Word  V  ->  (
x cyclShift  0 )  =  x )
12113ad2ant2 1018 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( x cyclShift  0 )  =  x )
13 0elfz 11782 . . . . . . . . 9  |-  ( N  e.  NN0  ->  0  e.  ( 0 ... N
) )
1413adantr 465 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
0  e.  ( 0 ... N ) )
15143ad2ant3 1019 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  0  e.  ( 0 ... N
) )
16 eqcom 2476 . . . . . . . 8  |-  ( ( x cyclShift  0 )  =  x  <->  x  =  (
x cyclShift  0 ) )
1716biimpi 194 . . . . . . 7  |-  ( ( x cyclShift  0 )  =  x  ->  x  =  ( x cyclShift  0 ) )
18 oveq2 6302 . . . . . . . . 9  |-  ( n  =  0  ->  (
x cyclShift  n )  =  ( x cyclShift  0 ) )
1918eqeq2d 2481 . . . . . . . 8  |-  ( n  =  0  ->  (
x  =  ( x cyclShift  n )  <->  x  =  ( x cyclShift  0 ) ) )
2019rspcev 3219 . . . . . . 7  |-  ( ( 0  e.  ( 0 ... N )  /\  x  =  ( x cyclShift  0 ) )  ->  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) )
2115, 17, 20syl2an 477 . . . . . 6  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) )  /\  ( x cyclShift  0 )  =  x )  ->  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) )
2212, 21mpdan 668 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) )
2310, 22syl 16 . . . 4  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) )
2423, 6eleq2s 2575 . . 3  |-  ( x  e.  W  ->  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) )
2524pm4.71i 632 . 2  |-  ( x  e.  W  <->  ( x  e.  W  /\  E. n  e.  ( 0 ... N
) x  =  ( x cyclShift  n ) ) )
264, 9, 253bitr4ri 278 1  |-  ( x  e.  W  <->  x  .~  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   class class class wbr 4452   {copab 4509   ` cfv 5593  (class class class)co 6294   0cc0 9502   NN0cn0 10805   ...cfz 11682   #chash 12383  Word cword 12510   cyclShift ccsh 12734   ClWWalksN cclwwlkn 24540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-fzo 11803  df-fl 11907  df-mod 11975  df-hash 12384  df-word 12518  df-concat 12520  df-substr 12522  df-csh 12735  df-clwwlk 24542  df-clwwlkn 24543
This theorem is referenced by:  erclwwlkn  24619
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