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Theorem eleclclwwlknlem1 30661
Description: Lemma 1 for eleclclwwlkn 30678. (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 30606 . . . . . . . 8  |-  ( Y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  Y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 Y )  =  N ) ) )
2 simpr 461 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  ( # `  Y )  =  N )  -> 
( # `  Y )  =  N )
32anim2i 569 . . . . . . . . 9  |-  ( ( Y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  Y
)  =  N ) )  ->  ( Y  e. Word  V  /\  ( # `  Y )  =  N ) )
433adant1 1006 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  Y  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  Y
)  =  N ) )  ->  ( Y  e. Word  V  /\  ( # `  Y )  =  N ) )
51, 4syl 16 . . . . . . 7  |-  ( Y  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( Y  e. Word  V  /\  ( # `  Y )  =  N ) )
6 erclwwlkn1.w . . . . . . 7  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
75, 6eleq2s 2562 . . . . . 6  |-  ( Y  e.  W  ->  ( Y  e. Word  V  /\  ( # `
 Y )  =  N ) )
87adantl 466 . . . . 5  |-  ( ( X  e.  W  /\  Y  e.  W )  ->  ( Y  e. Word  V  /\  ( # `  Y
)  =  N ) )
98adantl 466 . . . 4  |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W
) )  ->  ( Y  e. Word  V  /\  ( # `
 Y )  =  N ) )
109adantr 465 . . 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 457 . . . . 5  |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W
) )  ->  K  e.  ( 0 ... N
) )
1211adantr 465 . . . 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 457 . . . . 5  |-  ( ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  X  =  ( Y cyclShift  K ) )
1413adantl 466 . . . 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 461 . . . . 5  |-  ( ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  E. m  e.  ( 0 ... N
) Z  =  ( Y cyclShift  m ) )
1615adantl 466 . . . 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 1168 . . 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 cshwlemma1 30660 . . 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 60 . 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 434 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800   _Vcvv 3078   ` cfv 5529  (class class class)co 6203   0cc0 9397   NN0cn0 10694   ...cfz 11558   #chash 12224  Word cword 12343   cyclShift ccsh 12547   ClWWalksN cclwwlkn 30585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-fzo 11670  df-fl 11763  df-mod 11830  df-hash 12225  df-word 12351  df-concat 12353  df-substr 12355  df-csh 12548  df-clwwlk 30587  df-clwwlkn 30588
This theorem is referenced by:  eleclclwwlknlem2  30662
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