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Theorem clwwlknwwlkncl 30597
Description: Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
Assertion
Ref Expression
clwwlknwwlkncl  |-  ( ( N  e.  NN  /\  P  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  ( P concat  <" ( P `
 0 ) "> )  e.  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( lastS  `  w
)  =  ( w `
 0 ) } )
Distinct variable groups:    w, E    w, N    w, P    w, V

Proof of Theorem clwwlknwwlkncl
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwwlknimp 30574 . . . 4  |-  ( P  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
) )
2 clwwlknprop 30570 . . . 4  |-  ( P  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )
3 df-3an 967 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN ) )
43simplbi2 625 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  NN  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )
) )
543ad2ant1 1009 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  P
)  =  N ) )  ->  ( N  e.  NN  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN ) ) )
65adantl 466 . . . . . . 7  |-  ( ( ( ( P  e. Word  V  /\  ( # `  P
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )  ->  ( N  e.  NN  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN ) ) )
76imp 429 . . . . . 6  |-  ( ( ( ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
)  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )  /\  N  e.  NN )  ->  ( V  e. 
_V  /\  E  e.  _V  /\  N  e.  NN ) )
8 simpll1 1027 . . . . . 6  |-  ( ( ( ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
)  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )  /\  N  e.  NN )  ->  ( P  e. Word  V  /\  ( # `  P
)  =  N ) )
9 3simpc 987 . . . . . . 7  |-  ( ( ( P  e. Word  V  /\  ( # `  P
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  ->  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) )
109ad2antrr 725 . . . . . 6  |-  ( ( ( ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
)  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )  /\  N  e.  NN )  ->  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) )
117, 8, 103jca 1168 . . . . 5  |-  ( ( ( ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
)  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )  /\  N  e.  NN )  ->  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )  /\  ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  ( A. i  e.  (
0..^ ( N  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P ` 
0 ) }  e.  ran  E ) ) )
1211ex 434 . . . 4  |-  ( ( ( ( P  e. Word  V  /\  ( # `  P
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  /\  ( ( V  e.  _V  /\  E  e.  _V )  /\  P  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 P )  =  N ) ) )  ->  ( N  e.  NN  ->  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )  /\  ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  ( A. i  e.  (
0..^ ( N  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P ` 
0 ) }  e.  ran  E ) ) ) )
131, 2, 12syl2anc 661 . . 3  |-  ( P  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )  /\  ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  ( A. i  e.  (
0..^ ( N  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P ` 
0 ) }  e.  ran  E ) ) ) )
1413impcom 430 . 2  |-  ( ( N  e.  NN  /\  P  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )  /\  ( P  e. Word  V  /\  ( # `  P
)  =  N )  /\  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) ) )
15 eqid 2451 . . 3  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) }  =  { w  e.  (
( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }
1615clwwlkel 30590 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN )  /\  ( P  e. Word  V  /\  ( # `  P
)  =  N )  /\  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) )  ->  ( P concat  <" ( P ` 
0 ) "> )  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( lastS  `  w )  =  ( w ` 
0 ) } )
1714, 16syl 16 1  |-  ( ( N  e.  NN  /\  P  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  ( P concat  <" ( P `
 0 ) "> )  e.  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( lastS  `  w
)  =  ( w `
 0 ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2793   {crab 2797   _Vcvv 3065   {cpr 3974   ran crn 4936   ` cfv 5513  (class class class)co 6187   0cc0 9380   1c1 9381    + caddc 9383    - cmin 9693   NNcn 10420   NN0cn0 10677  ..^cfzo 11646   #chash 12201  Word cword 12320   lastS clsw 12321   concat cconcat 12322   <"cs1 12323   WWalksN cwwlkn 30447   ClWWalksN cclwwlkn 30549
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-map 7313  df-pm 7314  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-card 8207  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-hash 12202  df-word 12328  df-lsw 12329  df-concat 12330  df-s1 12331  df-wwlk 30448  df-wwlkn 30449  df-clwwlk 30551  df-clwwlkn 30552
This theorem is referenced by: (None)
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