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Theorem wwlknextbi 24398
Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
Assertion
Ref Expression
wwlknextbi  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  <-> 
T  e.  ( ( V WWalksN  E ) `  N
) ) )

Proof of Theorem wwlknextbi
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wwlknimp 24360 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E
) )
2 wwlknred 24396 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W substr  <.
0 ,  ( N  +  1 ) >.
)  e.  ( ( V WWalksN  E ) `  N
) ) )
32ad2antrr 725 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  e.  ( ( V WWalksN  E
) `  N )
) )
4 fveq2 5864 . . . . . . . . . . . . . . . . 17  |-  ( W  =  ( T concat  <" S "> )  ->  ( # `
 W )  =  ( # `  ( T concat  <" S "> ) ) )
54eqeq1d 2469 . . . . . . . . . . . . . . . 16  |-  ( W  =  ( T concat  <" S "> )  ->  (
( # `  W )  =  ( ( N  +  1 )  +  1 )  <->  ( # `  ( T concat  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
653ad2ant2 1018 . . . . . . . . . . . . . . 15  |-  ( ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( ( # `  W
)  =  ( ( N  +  1 )  +  1 )  <->  ( # `  ( T concat  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
76adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 W )  =  ( ( N  + 
1 )  +  1 )  <->  ( # `  ( T concat  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
8 s1cl 12571 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( S  e.  V  ->  <" S ">  e. Word  V )
98adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  S  e.  V )  ->  <" S ">  e. Word  V )
109anim2i 569 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( T  e. Word  V  /\  ( N  e.  NN0  /\  S  e.  V ) )  ->  ( T  e. Word  V  /\  <" S ">  e. Word  V )
)
1110ancoms 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( T  e. Word  V  /\  <" S ">  e. Word  V )
)
12 ccatlen 12553 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T  e. Word  V  /\  <" S ">  e. Word  V )  ->  ( # `
 ( T concat  <" S "> ) )  =  ( ( # `  T
)  +  ( # `  <" S "> ) ) )
1311, 12syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  ( T concat  <" S "> ) )  =  ( ( # `  T
)  +  ( # `  <" S "> ) ) )
1413eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T concat  <" S "> ) )  =  ( ( N  + 
1 )  +  1 )  <->  ( ( # `  T )  +  (
# `  <" S "> ) )  =  ( ( N  + 
1 )  +  1 ) ) )
15 s1len 12574 . . . . . . . . . . . . . . . . . . . 20  |-  ( # `  <" S "> )  =  1
1615a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  <" S "> )  =  1 )
1716oveq2d 6298 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 T )  +  ( # `  <" S "> )
)  =  ( (
# `  T )  +  1 ) )
1817eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( (
( # `  T )  +  ( # `  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  <->  ( ( # `
 T )  +  1 )  =  ( ( N  +  1 )  +  1 ) ) )
19 lencl 12522 . . . . . . . . . . . . . . . . . . . 20  |-  ( T  e. Word  V  ->  ( # `
 T )  e. 
NN0 )
2019nn0cnd 10850 . . . . . . . . . . . . . . . . . . 19  |-  ( T  e. Word  V  ->  ( # `
 T )  e.  CC )
2120adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  T
)  e.  CC )
22 peano2nn0 10832 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
2322nn0cnd 10850 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  CC )
2423ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( N  +  1 )  e.  CC )
25 ax-1cn 9546 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
2625a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  1  e.  CC )
2721, 24, 26addcan2d 9779 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( (
( # `  T )  +  1 )  =  ( ( N  + 
1 )  +  1 )  <->  ( # `  T
)  =  ( N  +  1 ) ) )
2814, 18, 273bitrd 279 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T concat  <" S "> ) )  =  ( ( N  + 
1 )  +  1 )  <->  ( # `  T
)  =  ( N  +  1 ) ) )
29 opeq2 4214 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  +  1 )  =  ( # `  T
)  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( # `  T
) >. )
3029eqcoms 2479 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  T )  =  ( N  + 
1 )  ->  <. 0 ,  ( N  + 
1 ) >.  =  <. 0 ,  ( # `  T
) >. )
3130oveq2d 6298 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  T )  =  ( N  + 
1 )  ->  (
( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  ( ( T concat  <" S "> ) substr  <. 0 ,  ( # `  T
) >. ) )
32 swrdccat1 12639 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T  e. Word  V  /\  <" S ">  e. Word  V )  ->  (
( T concat  <" S "> ) substr  <. 0 ,  ( # `  T
) >. )  =  T )
3311, 32syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( # `  T
) >. )  =  T )
3431, 33sylan9eqr 2530 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  T  e. Word  V )  /\  ( # `
 T )  =  ( N  +  1 ) )  ->  (
( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T )
3534ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 T )  =  ( N  +  1 )  ->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
3628, 35sylbid 215 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T concat  <" S "> ) )  =  ( ( N  + 
1 )  +  1 )  ->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
37363ad2antr1 1161 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 ( T concat  <" S "> ) )  =  ( ( N  + 
1 )  +  1 )  ->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
387, 37sylbid 215 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 W )  =  ( ( N  + 
1 )  +  1 )  ->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
3938imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  {
( lastS  `  T ) ,  S }  e.  ran  E ) )  /\  ( # `
 W )  =  ( ( N  + 
1 )  +  1 ) )  ->  (
( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T )
40 oveq1 6289 . . . . . . . . . . . . . . . 16  |-  ( W  =  ( T concat  <" S "> )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  (
( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. ) )
4140eqeq1d 2469 . . . . . . . . . . . . . . 15  |-  ( W  =  ( T concat  <" S "> )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. )  =  T  <-> 
( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
42413ad2ant2 1018 . . . . . . . . . . . . . 14  |-  ( ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  T  <->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 )
>. )  =  T
) )
4342adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  T  <->  ( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
4443adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  {
( lastS  `  T ) ,  S }  e.  ran  E ) )  /\  ( # `
 W )  =  ( ( N  + 
1 )  +  1 ) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. )  =  T  <-> 
( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
4539, 44mpbird 232 . . . . . . . . . . 11  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  {
( lastS  `  T ) ,  S }  e.  ran  E ) )  /\  ( # `
 W )  =  ( ( N  + 
1 )  +  1 ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  T
)
4645eleq1d 2536 . . . . . . . . . 10  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  {
( lastS  `  T ) ,  S }  e.  ran  E ) )  /\  ( # `
 W )  =  ( ( N  + 
1 )  +  1 ) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. )  e.  ( ( V WWalksN  E ) `  N )  <->  T  e.  ( ( V WWalksN  E
) `  N )
) )
4746biimpd 207 . . . . . . . . 9  |-  ( ( ( ( N  e. 
NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  {
( lastS  `  T ) ,  S }  e.  ran  E ) )  /\  ( # `
 W )  =  ( ( N  + 
1 )  +  1 ) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. )  e.  ( ( V WWalksN  E ) `  N )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) )
4847ex 434 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( # `
 W )  =  ( ( N  + 
1 )  +  1 )  ->  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  e.  (
( V WWalksN  E ) `  N )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
4948com23 78 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  e.  (
( V WWalksN  E ) `  N )  ->  (
( # `  W )  =  ( ( N  +  1 )  +  1 )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
503, 49syld 44 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( ( # `  W )  =  ( ( N  +  1 )  +  1 )  ->  T  e.  ( ( V WWalksN  E ) `  N ) ) ) )
5150com13 80 . . . . 5  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( (
( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
52513ad2ant2 1018 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( (
( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) ) )
531, 52mpcom 36 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( (
( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  T  e.  ( ( V WWalksN  E
) `  N )
) )
5453com12 31 . 2  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  T  e.  ( ( V WWalksN  E ) `  N ) ) )
55 wwlknext 24397 . . . . . . . . . . 11  |-  ( ( T  e.  ( ( V WWalksN  E ) `  N
)  /\  S  e.  V  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( T concat  <" S "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )
56 eleq1 2539 . . . . . . . . . . 11  |-  ( W  =  ( T concat  <" S "> )  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  <->  ( T concat  <" S "> )  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
5755, 56syl5ibrcom 222 . . . . . . . . . 10  |-  ( ( T  e.  ( ( V WWalksN  E ) `  N
)  /\  S  e.  V  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( W  =  ( T concat  <" S "> )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )
58573exp 1195 . . . . . . . . 9  |-  ( T  e.  ( ( V WWalksN  E ) `  N
)  ->  ( S  e.  V  ->  ( { ( lastS  `  T ) ,  S }  e.  ran  E  ->  ( W  =  ( T concat  <" S "> )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) ) )
5958com23 78 . . . . . . . 8  |-  ( T  e.  ( ( V WWalksN  E ) `  N
)  ->  ( {
( lastS  `  T ) ,  S }  e.  ran  E  ->  ( S  e.  V  ->  ( W  =  ( T concat  <" S "> )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) ) )
6059com14 88 . . . . . . 7  |-  ( W  =  ( T concat  <" S "> )  ->  ( { ( lastS  `  T ) ,  S }  e.  ran  E  ->  ( S  e.  V  ->  ( T  e.  ( ( V WWalksN  E ) `  N
)  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) ) )
6160imp 429 . . . . . 6  |-  ( ( W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( S  e.  V  ->  ( T  e.  ( ( V WWalksN  E ) `  N )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
62613adant1 1014 . . . . 5  |-  ( ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T
) ,  S }  e.  ran  E )  -> 
( S  e.  V  ->  ( T  e.  ( ( V WWalksN  E ) `  N )  ->  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
6362com12 31 . . . 4  |-  ( S  e.  V  ->  (
( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E )  ->  ( T  e.  ( ( V WWalksN  E
) `  N )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) ) )
6463adantl 466 . . 3  |-  ( ( N  e.  NN0  /\  S  e.  V )  ->  ( ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E )  ->  ( T  e.  ( ( V WWalksN  E
) `  N )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) ) )
6564imp 429 . 2  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( T  e.  ( ( V WWalksN  E
) `  N )  ->  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
6654, 65impbid 191 1  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  ( T  e. Word  V  /\  W  =  ( T concat  <" S "> )  /\  { ( lastS  `  T ) ,  S }  e.  ran  E ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  <-> 
T  e.  ( ( V WWalksN  E ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {cpr 4029   <.cop 4033   ran crn 5000   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491   NN0cn0 10791  ..^cfzo 11788   #chash 12367  Word cword 12494   lastS clsw 12495   concat cconcat 12496   <"cs1 12497   substr csubstr 12498   WWalksN cwwlkn 24351
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 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-card 8316  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-hash 12368  df-word 12502  df-lsw 12503  df-concat 12504  df-s1 12505  df-substr 12506  df-wwlk 24352  df-wwlkn 24353
This theorem is referenced by:  wwlkextwrd  24401
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