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Theorem erclwwlknref 30504
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 967 . . 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 2980 . . 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 30502 . . 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 30440 . . . . 5  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
11 cshw0 12436 . . . . . . 7  |-  ( x  e. Word  V  ->  (
x cyclShift  0 )  =  x )
12113ad2ant2 1010 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( x cyclShift  0 )  =  x )
13 0elfz 11488 . . . . . . . . 9  |-  ( N  e.  NN0  ->  0  e.  ( 0 ... N
) )
1413adantr 465 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
0  e.  ( 0 ... N ) )
15143ad2ant3 1011 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  0  e.  ( 0 ... N
) )
16 eqcom 2445 . . . . . . . 8  |-  ( ( x cyclShift  0 )  =  x  <->  x  =  (
x cyclShift  0 ) )
1716biimpi 194 . . . . . . 7  |-  ( ( x cyclShift  0 )  =  x  ->  x  =  ( x cyclShift  0 ) )
18 oveq2 6104 . . . . . . . . 9  |-  ( n  =  0  ->  (
x cyclShift  n )  =  ( x cyclShift  0 ) )
1918eqeq2d 2454 . . . . . . . 8  |-  ( n  =  0  ->  (
x  =  ( x cyclShift  n )  <->  x  =  ( x cyclShift  0 ) ) )
2019rspcev 3078 . . . . . . 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 2535 . . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2721   _Vcvv 2977   class class class wbr 4297   {copab 4354   ` cfv 5423  (class class class)co 6096   0cc0 9287   NN0cn0 10584   ...cfz 11442   #chash 12108  Word cword 12226   cyclShift ccsh 12430   ClWWalksN cclwwlkn 30419
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-hash 12109  df-word 12234  df-concat 12236  df-substr 12238  df-csh 12431  df-clwwlk 30421  df-clwwlkn 30422
This theorem is referenced by:  erclwwlkn  30507
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