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Theorem erclwwlknsym 24649
Description:  .~ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-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
erclwwlknsym  |-  ( x  .~  y  ->  y  .~  x )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    x, n, t, u    n, N    y, n, t, u, x    n, W
Allowed substitution hints:    .~ ( x, y, u, t, n)    E( x, y, n)    N( x, y)    V( x, y)    W( x, y)

Proof of Theorem erclwwlknsym
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 vex 3121 . 2  |-  x  e. 
_V
2 vex 3121 . 2  |-  y  e. 
_V
3 erclwwlkn.w . . . 4  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
4 erclwwlkn.r . . . 4  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
53, 4erclwwlkneqlen 24647 . . 3  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  ->  ( # `  x
)  =  ( # `  y ) ) )
63, 4erclwwlkneq 24646 . . . 4  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  <->  ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n
) ) ) )
7 simpl2 1000 . . . . . . 7  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  y  e.  W )
8 simpl1 999 . . . . . . 7  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  x  e.  W )
9 clwwlknprop 24595 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
10 eqcom 2476 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  x )  =  N  <->  N  =  ( # `
 x ) )
1110biimpi 194 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  x )  =  N  ->  N  =  ( # `  x
) )
1211adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  ->  N  =  ( # `  x
) )
13123ad2ant3 1019 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  N  =  ( # `  x ) )
149, 13syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  N  =  ( # `  x ) )
1514, 3eleq2s 2575 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  W  ->  N  =  ( # `  x
) )
1615adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  W  /\  y  e.  W )  ->  N  =  ( # `  x ) )
1716adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  ->  N  =  ( # `  x
) )
18 clwwlknprop 24595 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 y )  =  N ) ) )
1918simp2d 1009 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  y  e. Word  V )
2019, 3eleq2s 2575 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  W  ->  y  e. Word  V )
2120adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  W  /\  y  e.  W )  ->  y  e. Word  V )
2221adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  -> 
y  e. Word  V )
2322adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  y  e. Word  V
)
24 simprr 756 . . . . . . . . . . . . . . . . 17  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( # `  x
)  =  ( # `  y ) )
2523, 24cshwcshid 12775 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( ( n  e.  ( 0 ... ( # `  y
) )  /\  x  =  ( y cyclShift  n
) )  ->  E. m  e.  ( 0 ... ( # `
 x ) ) y  =  ( x cyclShift  m ) ) )
26 oveq2 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  x
)  ->  ( 0 ... N )  =  ( 0 ... ( # `
 x ) ) )
27 oveq2 6303 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( 0 ... ( # `  x
) )  =  ( 0 ... ( # `  y ) ) )
2827adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  -> 
( 0 ... ( # `
 x ) )  =  ( 0 ... ( # `  y
) ) )
2926, 28sylan9eq 2528 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( 0 ... N )  =  ( 0 ... ( # `  y ) ) )
3029eleq2d 2537 . . . . . . . . . . . . . . . . 17  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( n  e.  ( 0 ... N
)  <->  n  e.  (
0 ... ( # `  y
) ) ) )
3130anbi1d 704 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( ( n  e.  ( 0 ... N )  /\  x  =  ( y cyclShift  n
) )  <->  ( n  e.  ( 0 ... ( # `
 y ) )  /\  x  =  ( y cyclShift  n ) ) ) )
3226adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( 0 ... N )  =  ( 0 ... ( # `  x ) ) )
3332rexeqdv 3070 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m )  <->  E. m  e.  ( 0 ... ( # `
 x ) ) y  =  ( x cyclShift  m ) ) )
3425, 31, 333imtr4d 268 . . . . . . . . . . . . . . 15  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( ( n  e.  ( 0 ... N )  /\  x  =  ( y cyclShift  n
) )  ->  E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m ) ) )
3517, 34mpancom 669 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  -> 
( ( n  e.  ( 0 ... N
)  /\  x  =  ( y cyclShift  n ) )  ->  E. m  e.  ( 0 ... N ) y  =  ( x cyclShift  m ) ) )
3635expd 436 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  -> 
( n  e.  ( 0 ... N )  ->  ( x  =  ( y cyclShift  n )  ->  E. m  e.  ( 0 ... N ) y  =  ( x cyclShift  m ) ) ) )
3736rexlimdv 2957 . . . . . . . . . . . 12  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  -> 
( E. n  e.  ( 0 ... N
) x  =  ( y cyclShift  n )  ->  E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m ) ) )
3837ex 434 . . . . . . . . . . 11  |-  ( ( x  e.  W  /\  y  e.  W )  ->  ( ( # `  x
)  =  ( # `  y )  ->  ( E. n  e.  (
0 ... N ) x  =  ( y cyclShift  n
)  ->  E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m ) ) ) )
3938com23 78 . . . . . . . . . 10  |-  ( ( x  e.  W  /\  y  e.  W )  ->  ( E. n  e.  ( 0 ... N
) x  =  ( y cyclShift  n )  ->  (
( # `  x )  =  ( # `  y
)  ->  E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m ) ) ) )
40393impia 1193 . . . . . . . . 9  |-  ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n
) )  ->  (
( # `  x )  =  ( # `  y
)  ->  E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m ) ) )
4140imp 429 . . . . . . . 8  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  E. m  e.  ( 0 ... N
) y  =  ( x cyclShift  m ) )
42 oveq2 6303 . . . . . . . . . 10  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
4342eqeq2d 2481 . . . . . . . . 9  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
4443cbvrexv 3094 . . . . . . . 8  |-  ( E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
)  <->  E. m  e.  ( 0 ... N ) y  =  ( x cyclShift  m ) )
4541, 44sylibr 212 . . . . . . 7  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) )
467, 8, 453jca 1176 . . . . . 6  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  (
y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) )
473, 4erclwwlkneq 24646 . . . . . . 7  |-  ( ( y  e.  _V  /\  x  e.  _V )  ->  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
4847ancoms 453 . . . . . 6  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( y  .~  x  <->  ( y  e.  W  /\  x  e.  W  /\  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) ) ) )
4946, 48syl5ibr 221 . . . . 5  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N
) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  y  .~  x ) )
5049expd 436 . . . 4  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N
) x  =  ( y cyclShift  n ) )  -> 
( ( # `  x
)  =  ( # `  y )  ->  y  .~  x ) ) )
516, 50sylbid 215 . . 3  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  ->  ( ( # `  x
)  =  ( # `  y )  ->  y  .~  x ) ) )
525, 51mpdd 40 . 2  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  ->  y  .~  x ) )
531, 2, 52mp2an 672 1  |-  ( x  .~  y  ->  y  .~  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   class class class wbr 4453   {copab 4510   ` cfv 5594  (class class class)co 6295   0cc0 9504   NN0cn0 10807   ...cfz 11684   #chash 12385  Word cword 12515   cyclShift ccsh 12739   ClWWalksN cclwwlkn 24572
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-hash 12386  df-word 12523  df-concat 12525  df-substr 12527  df-csh 12740  df-clwwlk 24574  df-clwwlkn 24575
This theorem is referenced by:  erclwwlkn  24651  eclclwwlkn1  24655
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