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Theorem clwlkisclwwlk 24493
Description: A closed walk as word corresponds to a closed walk in an undirected graph. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
Assertion
Ref Expression
clwlkisclwwlk  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( E. f 
f ( V ClWalks  E
) P  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. )  e.  ( V ClWWalks  E ) ) ) )
Distinct variable groups:    f, E    P, f    f, V

Proof of Theorem clwlkisclwwlk
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwlkisclwwlklem0 24492 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( E. f
( ( f  e. Word  dom  E  /\  P :
( 0 ... ( # `
 f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) )  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( A. i  e.  ( 0..^ ( ( ( ( # `  P
)  -  1 )  -  0 )  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( P `  (
( # `  P )  -  2 ) ) ,  ( P ` 
0 ) }  e.  ran  E ) ) ) )
2 lencl 12528 . . . . . . . . . . . . . 14  |-  ( P  e. Word  V  ->  ( # `
 P )  e. 
NN0 )
3 ige2m1fz 11767 . . . . . . . . . . . . . 14  |-  ( ( ( # `  P
)  e.  NN0  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  1 )  e.  ( 0 ... ( # `
 P ) ) )
42, 3sylan 471 . . . . . . . . . . . . 13  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  1 )  e.  ( 0 ... ( # `
 P ) ) )
5 swrd0len 12612 . . . . . . . . . . . . 13  |-  ( ( P  e. Word  V  /\  ( ( # `  P
)  -  1 )  e.  ( 0 ... ( # `  P
) ) )  -> 
( # `  ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) )  =  ( ( # `  P
)  -  1 ) )
64, 5syldan 470 . . . . . . . . . . . 12  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  ( # `
 ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) )  =  ( ( # `  P
)  -  1 ) )
72nn0cnd 10854 . . . . . . . . . . . . . . . 16  |-  ( P  e. Word  V  ->  ( # `
 P )  e.  CC )
8 1cnd 9612 . . . . . . . . . . . . . . . 16  |-  ( P  e. Word  V  ->  1  e.  CC )
97, 8subcld 9930 . . . . . . . . . . . . . . 15  |-  ( P  e. Word  V  ->  (
( # `  P )  -  1 )  e.  CC )
109subid1d 9919 . . . . . . . . . . . . . 14  |-  ( P  e. Word  V  ->  (
( ( # `  P
)  -  1 )  -  0 )  =  ( ( # `  P
)  -  1 ) )
1110eqcomd 2475 . . . . . . . . . . . . 13  |-  ( P  e. Word  V  ->  (
( # `  P )  -  1 )  =  ( ( ( # `  P )  -  1 )  -  0 ) )
1211adantr 465 . . . . . . . . . . . 12  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  1 )  =  ( ( ( # `  P )  -  1 )  -  0 ) )
136, 12eqtrd 2508 . . . . . . . . . . 11  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  ( # `
 ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) )  =  ( ( ( # `  P )  -  1 )  -  0 ) )
1413oveq1d 6299 . . . . . . . . . 10  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( # `  ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) )  -  1 )  =  ( ( ( ( # `  P
)  -  1 )  -  0 )  - 
1 ) )
1514oveq2d 6300 . . . . . . . . 9  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) )  - 
1 ) )  =  ( 0..^ ( ( ( ( # `  P
)  -  1 )  -  0 )  - 
1 ) ) )
166oveq1d 6299 . . . . . . . . . . . . 13  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( # `  ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) )  -  1 )  =  ( ( ( # `  P
)  -  1 )  -  1 ) )
1716oveq2d 6300 . . . . . . . . . . . 12  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) )  - 
1 ) )  =  ( 0..^ ( ( ( # `  P
)  -  1 )  -  1 ) ) )
1817eleq2d 2537 . . . . . . . . . . 11  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) )  <->  i  e.  ( 0..^ ( ( (
# `  P )  -  1 )  - 
1 ) ) ) )
19 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  P  e. Word  V )
20 wrdlenge2n0 12542 . . . . . . . . . . . . . . . 16  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  P  =/=  (/) )
2120adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  P  =/=  (/) )
22 nn0z 10887 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  P )  e.  NN0  ->  ( # `  P
)  e.  ZZ )
23 peano2zm 10906 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  P )  e.  ZZ  ->  ( ( # `
 P )  - 
1 )  e.  ZZ )
2422, 23syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  P )  e.  NN0  ->  ( ( # `
 P )  - 
1 )  e.  ZZ )
252, 24syl 16 . . . . . . . . . . . . . . . . 17  |-  ( P  e. Word  V  ->  (
( # `  P )  -  1 )  e.  ZZ )
2625adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( # `  P )  -  1 )  e.  ZZ )
27 elfzom1elfzo 11852 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( # `  P
)  -  1 )  e.  ZZ  /\  i  e.  ( 0..^ ( ( ( # `  P
)  -  1 )  -  1 ) ) )  ->  i  e.  ( 0..^ ( ( # `  P )  -  1 ) ) )
2826, 27sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )
29 swrdtrcfv 12631 . . . . . . . . . . . . . . 15  |-  ( ( P  e. Word  V  /\  P  =/=  (/)  /\  i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  i )  =  ( P `  i ) )
3019, 21, 28, 29syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  (
( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  i )  =  ( P `  i ) )
312adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  ( # `
 P )  e. 
NN0 )
32 elfzom1elp1fzo 11851 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( # `  P
)  -  1 )  e.  ZZ  /\  i  e.  ( 0..^ ( ( ( # `  P
)  -  1 )  -  1 ) ) )  ->  ( i  +  1 )  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )
3324, 32sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  P
)  e.  NN0  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  (
i  +  1 )  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )
3431, 33sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  (
i  +  1 )  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) )
35 swrdtrcfv 12631 . . . . . . . . . . . . . . 15  |-  ( ( P  e. Word  V  /\  P  =/=  (/)  /\  ( i  +  1 )  e.  ( 0..^ ( (
# `  P )  -  1 ) ) )  ->  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) )  =  ( P `  ( i  +  1 ) ) )
3619, 21, 34, 35syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  (
( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) )  =  ( P `  ( i  +  1 ) ) )
3730, 36preq12d 4114 . . . . . . . . . . . . 13  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )
3837eleq1d 2536 . . . . . . . . . . . 12  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) ) )  ->  ( { ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) `  i ) ,  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) )
3938ex 434 . . . . . . . . . . 11  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
i  e.  ( 0..^ ( ( ( # `  P )  -  1 )  -  1 ) )  ->  ( {
( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) ) )
4018, 39sylbid 215 . . . . . . . . . 10  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) )  ->  ( { ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) `  i ) ,  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) ) )
4140imp 429 . . . . . . . . 9  |-  ( ( ( P  e. Word  V  /\  2  <_  ( # `  P ) )  /\  i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) )  -> 
( { ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  i ) ,  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E ) )
4215, 41raleqbidva 3074 . . . . . . . 8  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  ( A. i  e.  (
0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) )  - 
1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  <->  A. i  e.  ( 0..^ ( ( ( (
# `  P )  -  1 )  - 
0 )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E ) )
43 swrdtrcfvl 12638 . . . . . . . . . 10  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) )  =  ( P `  (
( # `  P )  -  2 ) ) )
44 swrdtrcfv0 12632 . . . . . . . . . 10  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  0 )  =  ( P ` 
0 ) )
4543, 44preq12d 4114 . . . . . . . . 9  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  =  {
( P `  (
( # `  P )  -  2 ) ) ,  ( P ` 
0 ) } )
4645eleq1d 2536 . . . . . . . 8  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  ( { ( lastS  `  ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E  <->  { ( P `  ( ( # `  P
)  -  2 ) ) ,  ( P `
 0 ) }  e.  ran  E ) )
4742, 46anbi12d 710 . . . . . . 7  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( A. i  e.  ( 0..^ ( (
# `  ( P substr  <.
0 ,  ( (
# `  P )  -  1 ) >.
) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E )  <->  ( A. i  e.  ( 0..^ ( ( ( ( # `  P
)  -  1 )  -  0 )  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( P `  (
( # `  P )  -  2 ) ) ,  ( P ` 
0 ) }  e.  ran  E ) ) )
4847bicomd 201 . . . . . 6  |-  ( ( P  e. Word  V  /\  2  <_  ( # `  P
) )  ->  (
( A. i  e.  ( 0..^ ( ( ( ( # `  P
)  -  1 )  -  0 )  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( P `  (
( # `  P )  -  2 ) ) ,  ( P ` 
0 ) }  e.  ran  E )  <->  ( A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
49483adant1 1014 . . . . 5  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( ( A. i  e.  ( 0..^ ( ( ( (
# `  P )  -  1 )  - 
0 )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( P `  ( (
# `  P )  -  2 ) ) ,  ( P ` 
0 ) }  e.  ran  E )  <->  ( A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
50 swrdcl 12609 . . . . . . 7  |-  ( P  e. Word  V  ->  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. )  e. Word  V )
51503ad2ant2 1018 . . . . . 6  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. )  e. Word  V
)
52513biant1d 1337 . . . . 5  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( ( A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E )  <->  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
5349, 52bitrd 253 . . . 4  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( ( A. i  e.  ( 0..^ ( ( ( (
# `  P )  -  1 )  - 
0 )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( P `  ( (
# `  P )  -  2 ) ) ,  ( P ` 
0 ) }  e.  ran  E )  <->  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
5453anbi2d 703 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( ( ( lastS  `  P )  =  ( P `  0 )  /\  ( A. i  e.  ( 0..^ ( ( ( ( # `  P
)  -  1 )  -  0 )  - 
1 ) ) { ( P `  i
) ,  ( P `
 ( i  +  1 ) ) }  e.  ran  E  /\  { ( P `  (
( # `  P )  -  2 ) ) ,  ( P ` 
0 ) }  e.  ran  E ) )  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) ) )
551, 54bitrd 253 . 2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( E. f
( ( f  e. Word  dom  E  /\  P :
( 0 ... ( # `
 f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) )  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) ) )
56 usgrav 24042 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
57 isclwlk 24460 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V ClWalks  E ) P  <->  ( (
f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) ) )
58 3an4anass 1219 . . . . . . 7  |-  ( ( ( f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) )  <->  ( (
f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
5957, 58syl6bbr 263 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V ClWalks  E ) P  <->  ( (
f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
6056, 59syl 16 . . . . 5  |-  ( V USGrph  E  ->  ( f ( V ClWalks  E ) P  <->  ( (
f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
6160adantr 465 . . . 4  |-  ( ( V USGrph  E  /\  P  e. Word  V )  ->  (
f ( V ClWalks  E
) P  <->  ( (
f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
) ) ( E `
 ( f `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
6261exbidv 1690 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V )  ->  ( E. f  f ( V ClWalks  E ) P  <->  E. f
( ( f  e. Word  dom  E  /\  P :
( 0 ... ( # `
 f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
63623adant3 1016 . 2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( E. f 
f ( V ClWalks  E
) P  <->  E. f
( ( f  e. Word  dom  E  /\  P :
( 0 ... ( # `
 f ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
64 isclwwlk 24472 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. )  e.  ( V ClWWalks  E )  <->  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
6556, 64syl 16 . . . 4  |-  ( V USGrph  E  ->  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
)  e.  ( V ClWWalks  E )  <->  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
66653ad2ant1 1017 . . 3  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
)  e.  ( V ClWWalks  E )  <->  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. )  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) )
6766anbi2d 703 . 2  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( ( ( lastS  `  P )  =  ( P `  0 )  /\  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. )  e.  ( V ClWWalks  E ) )  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( ( P substr  <. 0 ,  ( (
# `  P )  -  1 ) >.
)  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) )  -  1 ) ) { ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  i
) ,  ( ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  ( P substr  <. 0 ,  ( ( # `  P
)  -  1 )
>. ) ) ,  ( ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. ) `  0
) }  e.  ran  E ) ) ) )
6855, 63, 673bitr4d 285 1  |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `
 P ) )  ->  ( E. f 
f ( V ClWalks  E
) P  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( P substr  <. 0 ,  ( ( # `  P )  -  1 ) >. )  e.  ( V ClWWalks  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447   dom cdm 4999   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    + caddc 9495    <_ cle 9629    - cmin 9805   2c2 10585   NN0cn0 10795   ZZcz 10864   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   lastS clsw 12501   substr csubstr 12504   USGrph cusg 24034   ClWalks cclwlk 24451   ClWWalks cclwwlk 24452
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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-rmo 2822  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-lsw 12509  df-substr 12512  df-usgra 24037  df-wlk 24212  df-clwlk 24454  df-clwwlk 24455
This theorem is referenced by:  clwlkisclwwlk2  24494
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