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Theorem erclwwlkneqlen 25026
Description: If two classes are equivalent regarding  .~, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-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
erclwwlkneqlen  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  ->  ( # `  T
)  =  ( # `  U ) ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    T, n, t, u    U, n, t, u    n, W    n, X    n, Y
Allowed substitution hints:    .~ ( u, t, n)    E( n)    N( n)    X( u, t)    Y( u, t)

Proof of Theorem erclwwlkneqlen
StepHypRef Expression
1 erclwwlkn.w . . 3  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
2 erclwwlkn.r . . 3  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
31, 2erclwwlkneq 25025 . 2  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  <->  ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N ) T  =  ( U cyclShift  n ) ) ) )
4 fveq2 5848 . . . . . . . . 9  |-  ( T  =  ( U cyclShift  n )  ->  ( # `  T
)  =  ( # `  ( U cyclShift  n )
) )
5 clwwlknprop 24974 . . . . . . . . . . . . . 14  |-  ( U  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 U )  =  N ) ) )
65, 1eleq2s 2562 . . . . . . . . . . . . 13  |-  ( U  e.  W  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  U
)  =  N ) ) )
7 simpl2 998 . . . . . . . . . . . . . . 15  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  U
)  =  N ) )  /\  n  e.  ( 0 ... N
) )  ->  U  e. Word  V )
8 oveq2 6278 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  U
)  ->  ( 0 ... N )  =  ( 0 ... ( # `
 U ) ) )
98eleq2d 2524 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  U
)  ->  ( n  e.  ( 0 ... N
)  <->  n  e.  (
0 ... ( # `  U
) ) ) )
109eqcoms 2466 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  U )  =  N  ->  ( n  e.  ( 0 ... N )  <->  n  e.  ( 0 ... ( # `
 U ) ) ) )
1110adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( # `  U )  =  N )  -> 
( n  e.  ( 0 ... N )  <-> 
n  e.  ( 0 ... ( # `  U
) ) ) )
12113ad2ant3 1017 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  U
)  =  N ) )  ->  ( n  e.  ( 0 ... N
)  <->  n  e.  (
0 ... ( # `  U
) ) ) )
1312biimpa 482 . . . . . . . . . . . . . . 15  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  U
)  =  N ) )  /\  n  e.  ( 0 ... N
) )  ->  n  e.  ( 0 ... ( # `
 U ) ) )
147, 13jca 530 . . . . . . . . . . . . . 14  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  U
)  =  N ) )  /\  n  e.  ( 0 ... N
) )  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `
 U ) ) ) )
1514ex 432 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  U  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  U
)  =  N ) )  ->  ( n  e.  ( 0 ... N
)  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `
 U ) ) ) ) )
166, 15syl 16 . . . . . . . . . . . 12  |-  ( U  e.  W  ->  (
n  e.  ( 0 ... N )  -> 
( U  e. Word  V  /\  n  e.  (
0 ... ( # `  U
) ) ) ) )
1716ad2antlr 724 . . . . . . . . . . 11  |-  ( ( ( T  e.  W  /\  U  e.  W
)  /\  ( T  e.  X  /\  U  e.  Y ) )  -> 
( n  e.  ( 0 ... N )  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `  U ) ) ) ) )
1817imp 427 . . . . . . . . . 10  |-  ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  ->  ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `
 U ) ) ) )
19 elfzelz 11691 . . . . . . . . . . 11  |-  ( n  e.  ( 0 ... ( # `  U
) )  ->  n  e.  ZZ )
20 cshwlen 12761 . . . . . . . . . . 11  |-  ( ( U  e. Word  V  /\  n  e.  ZZ )  ->  ( # `  ( U cyclShift  n ) )  =  ( # `  U
) )
2119, 20sylan2 472 . . . . . . . . . 10  |-  ( ( U  e. Word  V  /\  n  e.  ( 0 ... ( # `  U
) ) )  -> 
( # `  ( U cyclShift  n ) )  =  ( # `  U
) )
2218, 21syl 16 . . . . . . . . 9  |-  ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  ->  ( # `
 ( U cyclShift  n ) )  =  ( # `  U ) )
234, 22sylan9eqr 2517 . . . . . . . 8  |-  ( ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  /\  T  =  ( U cyclShift  n ) )  ->  ( # `  T
)  =  ( # `  U ) )
2423ex 432 . . . . . . 7  |-  ( ( ( ( T  e.  W  /\  U  e.  W )  /\  ( T  e.  X  /\  U  e.  Y )
)  /\  n  e.  ( 0 ... N
) )  ->  ( T  =  ( U cyclShift  n )  ->  ( # `  T
)  =  ( # `  U ) ) )
2524rexlimdva 2946 . . . . . 6  |-  ( ( ( T  e.  W  /\  U  e.  W
)  /\  ( T  e.  X  /\  U  e.  Y ) )  -> 
( E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n )  ->  ( # `
 T )  =  ( # `  U
) ) )
2625ex 432 . . . . 5  |-  ( ( T  e.  W  /\  U  e.  W )  ->  ( ( T  e.  X  /\  U  e.  Y )  ->  ( E. n  e.  (
0 ... N ) T  =  ( U cyclShift  n )  ->  ( # `  T
)  =  ( # `  U ) ) ) )
2726com23 78 . . . 4  |-  ( ( T  e.  W  /\  U  e.  W )  ->  ( E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n )  ->  (
( T  e.  X  /\  U  e.  Y
)  ->  ( # `  T
)  =  ( # `  U ) ) ) )
28273impia 1191 . . 3  |-  ( ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N ) T  =  ( U cyclShift  n ) )  ->  ( ( T  e.  X  /\  U  e.  Y )  ->  ( # `  T
)  =  ( # `  U ) ) )
2928com12 31 . 2  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n ) )  -> 
( # `  T )  =  ( # `  U
) ) )
303, 29sylbid 215 1  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  ->  ( # `  T
)  =  ( # `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106   class class class wbr 4439   {copab 4496   ` cfv 5570  (class class class)co 6270   0cc0 9481   NN0cn0 10791   ZZcz 10860   ...cfz 11675   #chash 12387  Word cword 12518   cyclShift ccsh 12750   ClWWalksN cclwwlkn 24951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-hash 12388  df-word 12526  df-concat 12528  df-substr 12530  df-csh 12751  df-clwwlk 24953  df-clwwlkn 24954
This theorem is referenced by:  erclwwlknsym  25028  erclwwlkntr  25029
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