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Theorem erclwwlkref 24940
Description:  .~ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
Hypothesis
Ref Expression
erclwwlk.r  |-  .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  w
) ) u  =  ( w cyclShift  n )
) }
Assertion
Ref Expression
erclwwlkref  |-  ( x  e.  ( V ClWWalks  E )  <-> 
x  .~  x )
Distinct variable groups:    n, E, u, w    n, V, u, w    x, n, u, w
Allowed substitution hints:    .~ ( x, w, u, n)    E( x)    V( x)

Proof of Theorem erclwwlkref
StepHypRef Expression
1 anidm 644 . . . 4  |-  ( ( x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )
)  <->  x  e.  ( V ClWWalks  E ) )
21anbi1i 695 . . 3  |-  ( ( ( x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )
)  /\  E. n  e.  ( 0 ... ( # `
 x ) ) x  =  ( x cyclShift  n ) )  <->  ( x  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  x ) ) x  =  ( x cyclShift  n
) ) )
3 df-3an 975 . . 3  |-  ( ( x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  x ) ) x  =  ( x cyclShift  n
) )  <->  ( (
x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )
)  /\  E. n  e.  ( 0 ... ( # `
 x ) ) x  =  ( x cyclShift  n ) ) )
4 clwwlkprop 24897 . . . . 5  |-  ( x  e.  ( V ClWWalks  E )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  x  e. Word  V
) )
5 cshw0 12777 . . . . . . 7  |-  ( x  e. Word  V  ->  (
x cyclShift  0 )  =  x )
6 0nn0 10831 . . . . . . . . . 10  |-  0  e.  NN0
76a1i 11 . . . . . . . . 9  |-  ( x  e. Word  V  ->  0  e.  NN0 )
8 lencl 12569 . . . . . . . . 9  |-  ( x  e. Word  V  ->  ( # `
 x )  e. 
NN0 )
9 hashge0 12458 . . . . . . . . 9  |-  ( x  e. Word  V  ->  0  <_  ( # `  x
) )
10 elfz2nn0 11795 . . . . . . . . 9  |-  ( 0  e.  ( 0 ... ( # `  x
) )  <->  ( 0  e.  NN0  /\  ( # `
 x )  e. 
NN0  /\  0  <_  (
# `  x )
) )
117, 8, 9, 10syl3anbrc 1180 . . . . . . . 8  |-  ( x  e. Word  V  ->  0  e.  ( 0 ... ( # `
 x ) ) )
12 eqcom 2466 . . . . . . . . 9  |-  ( ( x cyclShift  0 )  =  x  <->  x  =  (
x cyclShift  0 ) )
1312biimpi 194 . . . . . . . 8  |-  ( ( x cyclShift  0 )  =  x  ->  x  =  ( x cyclShift  0 ) )
14 oveq2 6304 . . . . . . . . . 10  |-  ( n  =  0  ->  (
x cyclShift  n )  =  ( x cyclShift  0 ) )
1514eqeq2d 2471 . . . . . . . . 9  |-  ( n  =  0  ->  (
x  =  ( x cyclShift  n )  <->  x  =  ( x cyclShift  0 ) ) )
1615rspcev 3210 . . . . . . . 8  |-  ( ( 0  e.  ( 0 ... ( # `  x
) )  /\  x  =  ( x cyclShift  0
) )  ->  E. n  e.  ( 0 ... ( # `
 x ) ) x  =  ( x cyclShift  n ) )
1711, 13, 16syl2an 477 . . . . . . 7  |-  ( ( x  e. Word  V  /\  ( x cyclShift  0 )  =  x )  ->  E. n  e.  ( 0 ... ( # `
 x ) ) x  =  ( x cyclShift  n ) )
185, 17mpdan 668 . . . . . 6  |-  ( x  e. Word  V  ->  E. n  e.  ( 0 ... ( # `
 x ) ) x  =  ( x cyclShift  n ) )
19183ad2ant3 1019 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  x  e. Word  V )  ->  E. n  e.  ( 0 ... ( # `
 x ) ) x  =  ( x cyclShift  n ) )
204, 19syl 16 . . . 4  |-  ( x  e.  ( V ClWWalks  E )  ->  E. n  e.  ( 0 ... ( # `  x ) ) x  =  ( x cyclShift  n
) )
2120pm4.71i 632 . . 3  |-  ( x  e.  ( V ClWWalks  E )  <-> 
( x  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  x
) ) x  =  ( x cyclShift  n )
) )
222, 3, 213bitr4ri 278 . 2  |-  ( x  e.  ( V ClWWalks  E )  <-> 
( x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  x ) ) x  =  ( x cyclShift  n
) ) )
23 vex 3112 . . 3  |-  x  e. 
_V
24 erclwwlk.r . . . 4  |-  .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  w
) ) u  =  ( w cyclShift  n )
) }
2524erclwwlkeq 24938 . . 3  |-  ( ( x  e.  _V  /\  x  e.  _V )  ->  ( x  .~  x  <->  ( x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  x ) ) x  =  ( x cyclShift  n
) ) ) )
2623, 23, 25mp2an 672 . 2  |-  ( x  .~  x  <->  ( x  e.  ( V ClWWalks  E )  /\  x  e.  ( V ClWWalks  E )  /\  E. n  e.  ( 0 ... ( # `  x
) ) x  =  ( x cyclShift  n )
) )
2722, 26bitr4i 252 1  |-  ( x  e.  ( V ClWWalks  E )  <-> 
x  .~  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109   class class class wbr 4456   {copab 4514   ` cfv 5594  (class class class)co 6296   0cc0 9509    <_ cle 9646   NN0cn0 10816   ...cfz 11697   #chash 12408  Word cword 12538   cyclShift ccsh 12771   ClWWalks cclwwlk 24875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-hash 12409  df-word 12546  df-concat 12548  df-substr 12550  df-csh 12772  df-clwwlk 24878
This theorem is referenced by:  erclwwlk  24943
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