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Theorem isclwwlk 24441
Description: Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
isclwwlk  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( W  e.  ( V ClWWalks  E )  <->  ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
Distinct variable groups:    i, E    i, V    i, W
Allowed substitution hints:    X( i)    Y( i)

Proof of Theorem isclwwlk
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 clwwlk 24439 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWWalks  E )  =  { w  e. Word  V  |  ( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E ) } )
21eleq2d 2537 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( W  e.  ( V ClWWalks  E )  <->  W  e.  { w  e. Word  V  | 
( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E ) } ) )
3 fveq2 5864 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
43oveq1d 6297 . . . . . . 7  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
54oveq2d 6298 . . . . . 6  |-  ( w  =  W  ->  (
0..^ ( ( # `  w )  -  1 ) )  =  ( 0..^ ( ( # `  W )  -  1 ) ) )
6 fveq1 5863 . . . . . . . 8  |-  ( w  =  W  ->  (
w `  i )  =  ( W `  i ) )
7 fveq1 5863 . . . . . . . 8  |-  ( w  =  W  ->  (
w `  ( i  +  1 ) )  =  ( W `  ( i  +  1 ) ) )
86, 7preq12d 4114 . . . . . . 7  |-  ( w  =  W  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) } )
98eleq1d 2536 . . . . . 6  |-  ( w  =  W  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  i
) ,  ( W `
 ( i  +  1 ) ) }  e.  ran  E ) )
105, 9raleqbidv 3072 . . . . 5  |-  ( w  =  W  ->  ( A. i  e.  (
0..^ ( ( # `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )
11 fveq2 5864 . . . . . . 7  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
12 fveq1 5863 . . . . . . 7  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1311, 12preq12d 4114 . . . . . 6  |-  ( w  =  W  ->  { ( lastS  `  w ) ,  ( w `  0 ) }  =  { ( lastS  `  W ) ,  ( W `  0 ) } )
1413eleq1d 2536 . . . . 5  |-  ( w  =  W  ->  ( { ( lastS  `  w ) ,  ( w ` 
0 ) }  e.  ran  E  <->  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) )
1510, 14anbi12d 710 . . . 4  |-  ( w  =  W  ->  (
( A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E )  <-> 
( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
1615elrab 3261 . . 3  |-  ( W  e.  { w  e. Word  V  |  ( A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E ) }  <->  ( W  e. Word  V  /\  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
17 3anass 977 . . 3  |-  ( ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) { ( W `
 i ) ,  ( W `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E )  <-> 
( W  e. Word  V  /\  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
1816, 17bitr4i 252 . 2  |-  ( W  e.  { w  e. Word  V  |  ( A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E ) }  <->  ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) )
192, 18syl6bb 261 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( W  e.  ( V ClWWalks  E )  <->  ( W  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  W
) ,  ( W `
 0 ) }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   {cpr 4029   ran crn 5000   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801  ..^cfzo 11788   #chash 12367  Word cword 12494   lastS clsw 12495   ClWWalks cclwwlk 24421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-word 12502  df-clwwlk 24424
This theorem is referenced by:  clwwlkgt0  24444  clwwlkn2  24448  clwwlknimp  24449  clwlkisclwwlk  24462  clwwlkf  24467  clwwlkext2edg  24475  wwlkext2clwwlk  24476  clwwisshclww  24480  clwlkfclwwlk  24517  extwwlkfablem2  24752  numclwwlkovfel2  24757  numclwwlkovf2ex  24760  numclwwlkovgelim  24763
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