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Theorem wwlknextbi 30506
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 30470 . . . 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 30504 . . . . . . . 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 5800 . . . . . . . . . . . . . . . . 17  |-  ( W  =  ( T concat  <" S "> )  ->  ( # `
 W )  =  ( # `  ( T concat  <" S "> ) ) )
54eqeq1d 2456 . . . . . . . . . . . . . . . 16  |-  ( W  =  ( T concat  <" S "> )  ->  (
( # `  W )  =  ( ( N  +  1 )  +  1 )  <->  ( # `  ( T concat  <" S "> ) )  =  ( ( N  +  1 )  +  1 ) ) )
653ad2ant2 1010 . . . . . . . . . . . . . . 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 12412 . . . . . . . . . . . . . . . . . . . . . 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 12394 . . . . . . . . . . . . . . . . . . 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 2456 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 ( T concat  <" S "> ) )  =  ( ( N  + 
1 )  +  1 )  <->  ( ( # `  T )  +  (
# `  <" S "> ) )  =  ( ( N  + 
1 )  +  1 ) ) )
15 s1len 12415 . . . . . . . . . . . . . . . . . . . 20  |-  ( # `  <" S "> )  =  1
1615a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( # `  <" S "> )  =  1 )
1716oveq2d 6217 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( ( # `
 T )  +  ( # `  <" S "> )
)  =  ( (
# `  T )  +  1 ) )
1817eqeq1d 2456 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  ( (
( # `  T )  +  ( # `  <" S "> )
)  =  ( ( N  +  1 )  +  1 )  <->  ( ( # `
 T )  +  1 )  =  ( ( N  +  1 )  +  1 ) ) )
19 lencl 12368 . . . . . . . . . . . . . . . . . . . 20  |-  ( T  e. Word  V  ->  ( # `
 T )  e. 
NN0 )
2019nn0cnd 10750 . . . . . . . . . . . . . . . . . . 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 10732 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
2322nn0cnd 10750 . . . . . . . . . . . . . . . . . . 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 9452 . . . . . . . . . . . . . . . . . . 19  |-  1  e.  CC
2625a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  S  e.  V )  /\  T  e. Word  V
)  ->  1  e.  CC )
2721, 24, 26addcan2d 9685 . . . . . . . . . . . . . . . . 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 4169 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  +  1 )  =  ( # `  T
)  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( # `  T
) >. )
3029eqcoms 2466 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  T )  =  ( N  + 
1 )  ->  <. 0 ,  ( N  + 
1 ) >.  =  <. 0 ,  ( # `  T
) >. )
3130oveq2d 6217 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  T )  =  ( N  + 
1 )  ->  (
( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  ( ( T concat  <" S "> ) substr  <. 0 ,  ( # `  T
) >. ) )
32 swrdccat1 12470 . . . . . . . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . . . . 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 1153 . . . . . . . . . . . . . 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 6208 . . . . . . . . . . . . . . . 16  |-  ( W  =  ( T concat  <" S "> )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  (
( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. ) )
4140eqeq1d 2456 . . . . . . . . . . . . . . 15  |-  ( W  =  ( T concat  <" S "> )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. )  =  T  <-> 
( ( T concat  <" S "> ) substr  <. 0 ,  ( N  +  1 ) >. )  =  T ) )
42413ad2ant2 1010 . . . . . . . . . . . . . 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 2523 . . . . . . . . . 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 1010 . . . 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 30505 . . . . . . . . . . 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 2526 . . . . . . . . . . 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 1187 . . . . . . . . 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 1006 . . . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2799   {cpr 3988   <.cop 3992   ran crn 4950   ` cfv 5527  (class class class)co 6201   CCcc 9392   0cc0 9394   1c1 9395    + caddc 9397   NN0cn0 10691  ..^cfzo 11666   #chash 12221  Word cword 12340   lastS clsw 12341   concat cconcat 12342   <"cs1 12343   substr csubstr 12344   WWalksN cwwlkn 30461
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-lsw 12349  df-concat 12350  df-s1 12351  df-substr 12352  df-wwlk 30462  df-wwlkn 30463
This theorem is referenced by:  wwlkextwrd  30509
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