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Theorem wwlknredwwlkn0 24929
Description: For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.)
Assertion
Ref Expression
wwlknredwwlkn0  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  (
( W `  0
)  =  P  <->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) )
Distinct variable groups:    y, E    y, N    y, V    y, W    y, P

Proof of Theorem wwlknredwwlkn0
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wwlknredwwlkn 24928 . . . 4  |-  ( N  e.  NN0  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) )
21imp 427 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) )
3 simpl 455 . . . . . . . . 9  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y
)
43adantl 464 . . . . . . . 8  |-  ( ( ( ( ( W `
 0 )  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  /\  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y
)
5 fveq1 5847 . . . . . . . . . . . . . 14  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( y `  0
)  =  ( ( W substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
65eqcoms 2466 . . . . . . . . . . . . 13  |-  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( y `  0
)  =  ( ( W substr  <. 0 ,  ( N  +  1 )
>. ) `  0 ) )
76adantr 463 . . . . . . . . . . . 12  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
) )  ->  (
y `  0 )  =  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )
)
8 wwlknimp 24889 . . . . . . . . . . . . . . . . . 18  |-  ( 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
) )
9 nn0p1nn 10831 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
10 peano2nn0 10832 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 nn0re 10800 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  +  1 )  e.  NN0  ->  ( N  +  1 )  e.  RR )
12 lep1 10377 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  +  1 )  e.  RR  ->  ( N  +  1 )  <_  ( ( N  +  1 )  +  1 ) )
1310, 11, 123syl 20 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  1 )  <_ 
( ( N  + 
1 )  +  1 ) )
14 peano2nn0 10832 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e. 
NN0 )
1514nn0zd 10963 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  ZZ )
16 fznn 11751 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  +  1 )  +  1 )  e.  ZZ  ->  (
( N  +  1 )  e.  ( 1 ... ( ( N  +  1 )  +  1 ) )  <->  ( ( N  +  1 )  e.  NN  /\  ( N  +  1 )  <_  ( ( N  +  1 )  +  1 ) ) ) )
1710, 15, 163syl 20 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  e.  ( 1 ... ( ( N  + 
1 )  +  1 ) )  <->  ( ( N  +  1 )  e.  NN  /\  ( N  +  1 )  <_  ( ( N  +  1 )  +  1 ) ) ) )
189, 13, 17mpbir2and 920 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) )
19 oveq2 6278 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
1 ... ( # `  W
) )  =  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
2019eleq2d 2524 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  +  1 )  e.  ( 1 ... ( # `  W
) )  <->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) ) )
2118, 20syl5ibr 221 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
2221adantl 464 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
23 simpl 455 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  ->  W  e. Word  V )
2422, 23jctild 541 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) ) )
25243adant3 1014 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  ( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) ) )
268, 25syl 16 . . . . . . . . . . . . . . . . 17  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `
 W ) ) ) ) )
2726impcom 428 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
2827adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( ( W `  0
)  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )  -> 
( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
2928adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `
 W ) ) ) )
3029adantl 464 . . . . . . . . . . . . 13  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
) )  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
31 swrd0fv0 12656 . . . . . . . . . . . . 13  |-  ( ( W  e. Word  V  /\  ( N  +  1
)  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( W `  0
) )
3230, 31syl 16 . . . . . . . . . . . 12  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( W `
 0 ) )
33 simprll 761 . . . . . . . . . . . 12  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
) )  ->  ( W `  0 )  =  P )
347, 32, 333eqtrd 2499 . . . . . . . . . . 11  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
) )  ->  (
y `  0 )  =  P )
3534ex 432 . . . . . . . . . 10  |-  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( ( ( W `  0 )  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  ->  ( y `  0 )  =  P ) )
3635adantr 463 . . . . . . . . 9  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  (
( ( ( W `
 0 )  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  ->  ( y `  0 )  =  P ) )
3736impcom 428 . . . . . . . 8  |-  ( ( ( ( ( W `
 0 )  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  /\  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E ) )  ->  (
y `  0 )  =  P )
38 simpr 459 . . . . . . . . 9  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )
3938adantl 464 . . . . . . . 8  |-  ( ( ( ( ( W `
 0 )  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  /\  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E ) )  ->  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )
404, 37, 393jca 1174 . . . . . . 7  |-  ( ( ( ( ( W `
 0 )  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  /\  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E ) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E ) )
4140ex 432 . . . . . 6  |-  ( ( ( ( W ` 
0 )  =  P  /\  ( N  e. 
NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  /\  y  e.  ( ( V WWalksN  E
) `  N )
)  ->  ( (
( W substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E )  ->  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E
) ) )
4241reximdva 2929 . . . . 5  |-  ( ( ( W `  0
)  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )  -> 
( E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) )
4342ex 432 . . . 4  |-  ( ( W `  0 )  =  P  ->  (
( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( E. y  e.  (
( V WWalksN  E ) `  N ) ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E )  ->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) ) )
4443com13 80 . . 3  |-  ( E. y  e.  ( ( V WWalksN  E ) `  N
) ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E
)  ->  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) )  ->  ( ( W `  0 )  =  P  ->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) ) )
452, 44mpcom 36 . 2  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  (
( W `  0
)  =  P  ->  E. y  e.  (
( V WWalksN  E ) `  N ) ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E
) ) )
4627, 31syl 16 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( W `
 0 ) )
4746eqcomd 2462 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W `  0 )  =  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )
)
4847adantl 464 . . . . . . 7  |-  ( ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  ->  ( W `  0 )  =  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
) `  0 )
)
49 fveq1 5847 . . . . . . . . 9  |-  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( W substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( y `  0
) )
5049adantr 463 . . . . . . . 8  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( y `
 0 ) )
5150adantr 463 . . . . . . 7  |-  ( ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( y `
 0 ) )
52 simpr 459 . . . . . . . 8  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
y `  0 )  =  P )
5352adantr 463 . . . . . . 7  |-  ( ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  ->  (
y `  0 )  =  P )
5448, 51, 533eqtrd 2499 . . . . . 6  |-  ( ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )  ->  ( W `  0 )  =  P )
5554ex 432 . . . . 5  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W `  0 )  =  P ) )
56553adant3 1014 . . . 4  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  (
( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W `  0 )  =  P ) )
5756com12 31 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  (
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  ( W `  0 )  =  P ) )
5857rexlimdvw 2949 . 2  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( E. y  e.  (
( V WWalksN  E ) `  N ) ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E
)  ->  ( W `  0 )  =  P ) )
5945, 58impbid 191 1  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  (
( W `  0
)  =  P  <->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {cpr 4018   <.cop 4022   class class class wbr 4439   ran crn 4989   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618   NNcn 10531   NN0cn0 10791   ZZcz 10860   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   lastS clsw 12519   substr csubstr 12522   WWalksN cwwlkn 24880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-lsw 12527  df-substr 12530  df-wwlk 24881  df-wwlkn 24882
This theorem is referenced by:  rusgranumwlks  25158
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