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Theorem erclwwlknsym 30500
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 2975 . 2  |-  x  e. 
_V
2 vex 2975 . 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 30498 . . 3  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x  .~  y  ->  ( # `  x
)  =  ( # `  y ) ) )
63, 4erclwwlkneq 30497 . . . 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 992 . . . . . . 7  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  y  e.  W )
8 simpl1 991 . . . . . . 7  |-  ( ( ( x  e.  W  /\  y  e.  W  /\  E. n  e.  ( 0 ... N ) x  =  ( y cyclShift  n ) )  /\  ( # `  x )  =  ( # `  y
) )  ->  x  e.  W )
9 clwwlknprop 30435 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
10 eqcom 2445 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  x )  =  N  <->  N  =  ( # `
 x ) )
1110biimpi 194 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  x )  =  N  ->  N  =  ( # `  x
) )
1211adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  ->  N  =  ( # `  x
) )
13123ad2ant3 1011 . . . . . . . . . . . . . . . . . . 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 2535 . . . . . . . . . . . . . . . . 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 30435 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 y )  =  N ) ) )
1918simp2d 1001 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  y  e. Word  V )
2019, 3eleq2s 2535 . . . . . . . . . . . . . . . . . . . 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, 24erclwwlksym0 30478 . . . . . . . . . . . . . . . 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 6099 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  x
)  ->  ( 0 ... N )  =  ( 0 ... ( # `
 x ) ) )
27 oveq2 6099 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  ( # `  y
)  ->  ( 0 ... ( # `  x
) )  =  ( 0 ... ( # `  y ) ) )
2827adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) )  -> 
( 0 ... ( # `
 x ) )  =  ( 0 ... ( # `  y
) ) )
2926, 28sylan9eq 2495 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  =  ( # `  x )  /\  (
( x  e.  W  /\  y  e.  W
)  /\  ( # `  x
)  =  ( # `  y ) ) )  ->  ( 0 ... N )  =  ( 0 ... ( # `  y ) ) )
3029eleq2d 2510 . . . . . . . . . . . . . . . . 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 2924 . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 1184 . . . . . . . . 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 6099 . . . . . . . . . 10  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
4342eqeq2d 2454 . . . . . . . . 9  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
4443cbvrexv 2948 . . . . . . . 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 1168 . . . . . 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 30497 . . . . . . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972   class class class wbr 4292   {copab 4349   ` cfv 5418  (class class class)co 6091   0cc0 9282   NN0cn0 10579   ...cfz 11437   #chash 12103  Word cword 12221   cyclShift ccsh 12425   ClWWalksN cclwwlkn 30414
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-hash 12104  df-word 12229  df-concat 12231  df-substr 12233  df-csh 12426  df-clwwlk 30416  df-clwwlkn 30417
This theorem is referenced by:  erclwwlkn  30502  eclclwwlkn1  30506
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