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Definition df-clwwlk 24877
Description: Define the set of all Closed Walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlk 24876. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk.

Notice that by this definition, a single vertex cannot be represented as closed walk, since the word <" v "> with vertex v represents the walk "vv", which is a (closed) walk of length 1 (if there is an edge/loop from v to v). Therefore, a closed walk corresponds to a closed walk as word in an undirected graph only for walks of length at least 1, see clwlkisclwwlk2 24916. (Contributed by Alexander van der Vekens, 20-Mar-2018.)

Assertion
Ref Expression
df-clwwlk  |- ClWWalks  =  ( v  e.  _V , 
e  e.  _V  |->  { 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 group:    e, i, v, w

Detailed syntax breakdown of Definition df-clwwlk
StepHypRef Expression
1 cclwwlk 24874 . 2  class ClWWalks
2 vv . . 3  setvar  v
3 ve . . 3  setvar  e
4 cvv 3109 . . 3  class  _V
5 vi . . . . . . . . . 10  setvar  i
65cv 1394 . . . . . . . . 9  class  i
7 vw . . . . . . . . . 10  setvar  w
87cv 1394 . . . . . . . . 9  class  w
96, 8cfv 5594 . . . . . . . 8  class  ( w `
 i )
10 c1 9510 . . . . . . . . . 10  class  1
11 caddc 9512 . . . . . . . . . 10  class  +
126, 10, 11co 6296 . . . . . . . . 9  class  ( i  +  1 )
1312, 8cfv 5594 . . . . . . . 8  class  ( w `
 ( i  +  1 ) )
149, 13cpr 4034 . . . . . . 7  class  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }
153cv 1394 . . . . . . . 8  class  e
1615crn 5009 . . . . . . 7  class  ran  e
1714, 16wcel 1819 . . . . . 6  wff  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  e
18 cc0 9509 . . . . . . 7  class  0
19 chash 12407 . . . . . . . . 9  class  #
208, 19cfv 5594 . . . . . . . 8  class  ( # `  w )
21 cmin 9824 . . . . . . . 8  class  -
2220, 10, 21co 6296 . . . . . . 7  class  ( (
# `  w )  -  1 )
23 cfzo 11820 . . . . . . 7  class ..^
2418, 22, 23co 6296 . . . . . 6  class  ( 0..^ ( ( # `  w
)  -  1 ) )
2517, 5, 24wral 2807 . . . . 5  wff  A. i  e.  ( 0..^ ( (
# `  w )  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  e
26 clsw 12538 . . . . . . . 8  class lastS
278, 26cfv 5594 . . . . . . 7  class  ( lastS  `  w
)
2818, 8cfv 5594 . . . . . . 7  class  ( w `
 0 )
2927, 28cpr 4034 . . . . . 6  class  { ( lastS  `  w ) ,  ( w `  0 ) }
3029, 16wcel 1819 . . . . 5  wff  { ( lastS  `  w ) ,  ( w `  0 ) }  e.  ran  e
3125, 30wa 369 . . . 4  wff  ( A. i  e.  ( 0..^ ( ( # `  w
)  -  1 ) ) { ( w `
 i ) ,  ( w `  (
i  +  1 ) ) }  e.  ran  e  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  e )
322cv 1394 . . . . 5  class  v
3332cword 12537 . . . 4  class Word  v
3431, 7, 33crab 2811 . . 3  class  { 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 ) }
352, 3, 4, 4, 34cmpt2 6298 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { 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 ) } )
361, 35wceq 1395 1  wff ClWWalks  =  ( v  e.  _V , 
e  e.  _V  |->  { 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
This definition is referenced by:  clwwlk  24892  clwwlkprop  24896
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