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Theorem wwlknredwwlkn0 30508
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 30507 . . . 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 429 . . 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 457 . . . . . . . . 9  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y
)
43adantl 466 . . . . . . . 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 5799 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 30470 . . . . . . . . . . . . . . . . . 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 10731 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
10 peano2nn0 10732 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 nn0re 10700 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  +  1 )  e.  NN0  ->  ( N  +  1 )  e.  RR )
12 lep1 10280 . . . . . . . . . . . . . . . . . . . . . . . 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 10732 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e. 
NN0 )
1514nn0zd 10857 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  ZZ )
16 fznn 11644 . . . . . . . . . . . . . . . . . . . . . . . 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 913 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) )
19 oveq2 6209 . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
23 simpl 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  ->  W  e. Word  V )
2422, 23jctild 543 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) ) )
25243adant3 1008 . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
2827adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( W `  0
)  =  P  /\  ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )  -> 
( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
2928adantr 465 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 12455 . . . . . . . . . . . . 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 434 . . . . . . . . . 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 465 . . . . . . . . 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 430 . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E )
3938adantl 466 . . . . . . . 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 1168 . . . . . . 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 434 . . . . . 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 2934 . . . . 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 434 . . . 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 466 . . . . . . 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 5799 . . . . . . . . 9  |-  ( ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  y  ->  ( ( W substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( y `  0
) )
5049adantr 465 . . . . . . . 8  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
( W substr  <. 0 ,  ( N  +  1 ) >. ) `  0
)  =  ( y `
 0 ) )
5150adantr 465 . . . . . . 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 461 . . . . . . . 8  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
y `  0 )  =  P )
5352adantr 465 . . . . . . 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 434 . . . . 5  |-  ( ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  (
y `  0 )  =  P )  ->  (
( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W `  0 )  =  P ) )
56553adant3 1008 . . . 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 2950 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   {cpr 3988   <.cop 3992   class class class wbr 4401   ran crn 4950   ` cfv 5527  (class class class)co 6201   RRcr 9393   0cc0 9394   1c1 9395    + caddc 9397    <_ cle 9531   NNcn 10434   NN0cn0 10691   ZZcz 10758   ...cfz 11555  ..^cfzo 11666   #chash 12221  Word cword 12340   lastS clsw 12341   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-substr 12352  df-wwlk 30462  df-wwlkn 30463
This theorem is referenced by:  rusgranumwlks  30723
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