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Theorem eleclclwwlknlem1 25234
Description: Lemma 1 for eleclclwwlkn 25250. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
Hypothesis
Ref Expression
erclwwlkn1.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
Assertion
Ref Expression
eleclclwwlknlem1  |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W
) )  ->  (
( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  E. n  e.  ( 0 ... N
) Z  =  ( X cyclShift  n ) ) )
Distinct variable groups:    m, n, K    m, N, n    m, V, n    m, X, n   
m, Y, n    m, Z, n
Allowed substitution hints:    E( m, n)    W( m, n)

Proof of Theorem eleclclwwlknlem1
StepHypRef Expression
1 clwwlknprop 25189 . . . . . . . 8  |-  ( Y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  Y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 Y )  =  N ) ) )
2 simpr 459 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  ( # `  Y )  =  N )  -> 
( # `  Y )  =  N )
32anim2i 567 . . . . . . . . 9  |-  ( ( Y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  Y
)  =  N ) )  ->  ( Y  e. Word  V  /\  ( # `  Y )  =  N ) )
433adant1 1015 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  Y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  Y
)  =  N ) )  ->  ( Y  e. Word  V  /\  ( # `  Y )  =  N ) )
51, 4syl 17 . . . . . . 7  |-  ( Y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( Y  e. Word  V  /\  ( # `  Y )  =  N ) )
6 erclwwlkn1.w . . . . . . 7  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
75, 6eleq2s 2510 . . . . . 6  |-  ( Y  e.  W  ->  ( Y  e. Word  V  /\  ( # `
 Y )  =  N ) )
87adantl 464 . . . . 5  |-  ( ( X  e.  W  /\  Y  e.  W )  ->  ( Y  e. Word  V  /\  ( # `  Y
)  =  N ) )
98adantl 464 . . . 4  |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W
) )  ->  ( Y  e. Word  V  /\  ( # `
 Y )  =  N ) )
109adantr 463 . . 3  |-  ( ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  /\  ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) ) )  ->  ( Y  e. Word  V  /\  ( # `
 Y )  =  N ) )
11 simpl 455 . . . . 5  |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W
) )  ->  K  e.  ( 0 ... N
) )
1211adantr 463 . . . 4  |-  ( ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  /\  ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) ) )  ->  K  e.  ( 0 ... N
) )
13 simpl 455 . . . . 5  |-  ( ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  X  =  ( Y cyclShift  K ) )
1413adantl 464 . . . 4  |-  ( ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  /\  ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) ) )  ->  X  =  ( Y cyclShift  K ) )
15 simpr 459 . . . . 5  |-  ( ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  E. m  e.  ( 0 ... N
) Z  =  ( Y cyclShift  m ) )
1615adantl 464 . . . 4  |-  ( ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  /\  ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) ) )  ->  E. m  e.  ( 0 ... N
) Z  =  ( Y cyclShift  m ) )
1712, 14, 163jca 1177 . . 3  |-  ( ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  /\  ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) ) )  ->  ( K  e.  ( 0 ... N )  /\  X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N
) Z  =  ( Y cyclShift  m ) ) )
18 2cshwcshw 12849 . . 3  |-  ( ( Y  e. Word  V  /\  ( # `  Y )  =  N )  -> 
( ( K  e.  ( 0 ... N
)  /\  X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  E. n  e.  ( 0 ... N
) Z  =  ( X cyclShift  n ) ) )
1910, 17, 18sylc 59 . 2  |-  ( ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  /\  ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) ) )  ->  E. n  e.  ( 0 ... N
) Z  =  ( X cyclShift  n ) )
2019ex 432 1  |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W
) )  ->  (
( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  E. n  e.  ( 0 ... N
) Z  =  ( X cyclShift  n ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059   ` cfv 5569  (class class class)co 6278   0cc0 9522   NN0cn0 10836   ...cfz 11726   #chash 12452  Word cword 12583   cyclShift ccsh 12815   ClWWalksN cclwwlkn 25166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-hash 12453  df-word 12591  df-concat 12593  df-substr 12595  df-csh 12816  df-clwwlk 25168  df-clwwlkn 25169
This theorem is referenced by:  eleclclwwlknlem2  25235
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