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Theorem wwlknredwwlkn 24388
Description: For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.)
Assertion
Ref Expression
wwlknredwwlkn  |-  ( 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 ) ) )
Distinct variable groups:    y, E    y, N    y, V    y, W

Proof of Theorem wwlknredwwlkn
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 eqidd 2461 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  ( W substr  <. 0 ,  ( N  +  1 )
>. ) )
2 wwlknimp 24349 . . . 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
) )
3 simprl 755 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  W  e. Word  V
)
4 peano2nn0 10825 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5 peano2nn0 10825 . . . . . . . . . . . . 13  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e. 
NN0 )
64, 5syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e. 
NN0 )
7 id 22 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e. 
NN0 )
8 nn0p1nn 10824 . . . . . . . . . . . . . 14  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  NN )
94, 8syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  NN )
10 nn0re 10793 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  N  e.  RR )
11 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  N  e.  RR )
12 peano2re 9741 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
13 peano2re 9741 . . . . . . . . . . . . . . . . 17  |-  ( ( N  +  1 )  e.  RR  ->  (
( N  +  1 )  +  1 )  e.  RR )
1412, 13syl 16 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  (
( N  +  1 )  +  1 )  e.  RR )
1511, 12, 143jca 1171 . . . . . . . . . . . . . . 15  |-  ( N  e.  RR  ->  ( N  e.  RR  /\  ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  +  1 )  e.  RR ) )
1610, 15syl 16 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( N  e.  RR  /\  ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  +  1 )  e.  RR ) )
1710ltp1d 10465 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  < 
( N  +  1 ) )
18 nn0re 10793 . . . . . . . . . . . . . . . 16  |-  ( ( N  +  1 )  e.  NN0  ->  ( N  +  1 )  e.  RR )
194, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  RR )
2019ltp1d 10465 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( N  +  1 )  < 
( ( N  + 
1 )  +  1 ) )
21 lttr 9650 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  RR  /\  ( N  +  1
)  e.  RR  /\  ( ( N  + 
1 )  +  1 )  e.  RR )  ->  ( ( N  <  ( N  + 
1 )  /\  ( N  +  1 )  <  ( ( N  +  1 )  +  1 ) )  ->  N  <  ( ( N  +  1 )  +  1 ) ) )
2221imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  RR  /\  ( N  +  1 )  e.  RR  /\  ( ( N  + 
1 )  +  1 )  e.  RR )  /\  ( N  < 
( N  +  1 )  /\  ( N  +  1 )  < 
( ( N  + 
1 )  +  1 ) ) )  ->  N  <  ( ( N  +  1 )  +  1 ) )
2316, 17, 20, 22syl12anc 1221 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  < 
( ( N  + 
1 )  +  1 ) )
24 elfzo0 11820 . . . . . . . . . . . . 13  |-  ( N  e.  ( 0..^ ( ( N  +  1 )  +  1 ) )  <->  ( N  e. 
NN0  /\  ( ( N  +  1 )  +  1 )  e.  NN  /\  N  < 
( ( N  + 
1 )  +  1 ) ) )
257, 9, 23, 24syl3anbrc 1175 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( ( N  +  1 )  +  1 ) ) )
26 fargshiftlem 24296 . . . . . . . . . . . 12  |-  ( ( ( ( N  + 
1 )  +  1 )  e.  NN0  /\  N  e.  ( 0..^ ( ( N  + 
1 )  +  1 ) ) )  -> 
( N  +  1 )  e.  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
276, 25, 26syl2anc 661 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) )
2827adantr 465 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  ( N  + 
1 )  e.  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
29 oveq2 6283 . . . . . . . . . . . . 13  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
1 ... ( # `  W
) )  =  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
3029eleq2d 2530 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  +  1 )  e.  ( 1 ... ( # `  W
) )  <->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) ) )
3130adantl 466 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  + 
1 )  e.  ( 1 ... ( # `  W ) )  <->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) ) )
3231adantl 466 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  ( ( N  +  1 )  e.  ( 1 ... ( # `
 W ) )  <-> 
( N  +  1 )  e.  ( 1 ... ( ( N  +  1 )  +  1 ) ) ) )
3328, 32mpbird 232 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  ( N  + 
1 )  e.  ( 1 ... ( # `  W ) ) )
343, 33jca 532 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  ( W  e. Word  V  /\  ( N  + 
1 )  e.  ( 1 ... ( # `  W ) ) ) )
35343adantr3 1152 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  -> 
( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
36 swrd0fvlsw 12620 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( N  +  1
)  e.  ( 1 ... ( # `  W
) ) )  -> 
( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
)  =  ( W `
 ( ( N  +  1 )  - 
1 ) ) )
3735, 36syl 16 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  -> 
( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
)  =  ( W `
 ( ( N  +  1 )  - 
1 ) ) )
38 lsw 12537 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
39383ad2ant1 1012 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
4039adantl 466 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  -> 
( lastS  `  W )  =  ( W `  (
( # `  W )  -  1 ) ) )
4137, 40preq12d 4107 . . . . 5  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  W ) }  =  { ( W `  ( ( N  + 
1 )  -  1 ) ) ,  ( W `  ( (
# `  W )  -  1 ) ) } )
42 oveq1 6282 . . . . . . . . . . 11  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( # `  W )  -  1 )  =  ( ( ( N  +  1 )  +  1 )  -  1 ) )
43423ad2ant2 1013 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  (
( # `  W )  -  1 )  =  ( ( ( N  +  1 )  +  1 )  -  1 ) )
4443adantl 466 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  -> 
( ( # `  W
)  -  1 )  =  ( ( ( N  +  1 )  +  1 )  - 
1 ) )
4544fveq2d 5861 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  -> 
( W `  (
( # `  W )  -  1 ) )  =  ( W `  ( ( ( N  +  1 )  +  1 )  -  1 ) ) )
4645preq2d 4106 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( W `  ( ( N  + 
1 )  -  1 ) ) ,  ( W `  ( (
# `  W )  -  1 ) ) }  =  { ( W `  ( ( N  +  1 )  -  1 ) ) ,  ( W `  ( ( ( N  +  1 )  +  1 )  -  1 ) ) } )
47 nn0cn 10794 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
48 ax-1cn 9539 . . . . . . . . . . . 12  |-  1  e.  CC
4948a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  e.  CC )
5047, 49pncand 9920 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  -  1 )  =  N )
5150fveq2d 5861 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 ( ( N  +  1 )  - 
1 ) )  =  ( W `  N
) )
524nn0cnd 10843 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  CC )
5352, 49pncand 9920 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( ( ( N  +  1 )  +  1 )  -  1 )  =  ( N  +  1 ) )
5453fveq2d 5861 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 ( ( ( N  +  1 )  +  1 )  - 
1 ) )  =  ( W `  ( N  +  1 ) ) )
5551, 54preq12d 4107 . . . . . . . 8  |-  ( N  e.  NN0  ->  { ( W `  ( ( N  +  1 )  -  1 ) ) ,  ( W `  ( ( ( N  +  1 )  +  1 )  -  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1 ) ) } )
5655adantr 465 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( W `  ( ( N  + 
1 )  -  1 ) ) ,  ( W `  ( ( ( N  +  1 )  +  1 )  -  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1
) ) } )
5746, 56eqtrd 2501 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( W `  ( ( N  + 
1 )  -  1 ) ) ,  ( W `  ( (
# `  W )  -  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1
) ) } )
58 fzonn0p1 11849 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( N  +  1 ) ) )
59 fveq2 5857 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  ( W `  i )  =  ( W `  N ) )
60 oveq1 6282 . . . . . . . . . . . . . 14  |-  ( i  =  N  ->  (
i  +  1 )  =  ( N  + 
1 ) )
6160fveq2d 5861 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  ( W `  ( i  +  1 ) )  =  ( W `  ( N  +  1
) ) )
6259, 61preq12d 4107 . . . . . . . . . . . 12  |-  ( i  =  N  ->  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1 ) ) } )
6362eleq1d 2529 . . . . . . . . . . 11  |-  ( i  =  N  ->  ( { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  N
) ,  ( W `
 ( N  + 
1 ) ) }  e.  ran  E ) )
6463rspcv 3203 . . . . . . . . . 10  |-  ( N  e.  ( 0..^ ( N  +  1 ) )  ->  ( A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  ->  { ( W `  N ) ,  ( W `  ( N  +  1
) ) }  e.  ran  E ) )
6558, 64syl 16 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  ->  { ( W `  N ) ,  ( W `  ( N  +  1
) ) }  e.  ran  E ) )
6665com12 31 . . . . . . . 8  |-  ( A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  ->  ( N  e.  NN0  ->  { ( W `  N ) ,  ( W `  ( N  +  1
) ) }  e.  ran  E ) )
67663ad2ant3 1014 . . . . . . 7  |-  ( ( 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 `  N ) ,  ( W `  ( N  +  1
) ) }  e.  ran  E ) )
6867impcom 430 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( W `  N ) ,  ( W `  ( N  +  1 ) ) }  e.  ran  E
)
6957, 68eqeltrd 2548 . . . . 5  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( W `  ( ( N  + 
1 )  -  1 ) ) ,  ( W `  ( (
# `  W )  -  1 ) ) }  e.  ran  E
)
7041, 69eqeltrd 2548 . . . 4  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E ) )  ->  { ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  W ) }  e.  ran  E )
712, 70sylan2 474 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  { ( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) }  e.  ran  E )
72 wwlknred 24385 . . . . 5  |-  ( N  e.  NN0  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W substr  <.
0 ,  ( N  +  1 ) >.
)  e.  ( ( V WWalksN  E ) `  N
) ) )
7372imp 429 . . . 4  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  e.  (
( V WWalksN  E ) `  N ) )
74 eqeq2 2475 . . . . . 6  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  <->  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) )
75 fveq2 5857 . . . . . . . 8  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( lastS  `  y )  =  ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 ) >.
) ) )
7675preq1d 4105 . . . . . . 7  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  { ( lastS  `  y
) ,  ( lastS  `  W
) }  =  {
( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) } )
7776eleq1d 2529 . . . . . 6  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E  <->  { ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 ) >.
) ) ,  ( lastS  `  W ) }  e.  ran  E ) )
7874, 77anbi12d 710 . . . . 5  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E
)  <->  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( W substr  <. 0 ,  ( N  +  1 ) >.
)  /\  { ( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) }  e.  ran  E ) ) )
7978adantl 466 . . . 4  |-  ( ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  /\  y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
)  ->  ( (
( W substr  <. 0 ,  ( N  +  1 ) >. )  =  y  /\  { ( lastS  `  y
) ,  ( lastS  `  W
) }  e.  ran  E )  <->  ( ( W substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( W substr  <. 0 ,  ( N  +  1 ) >.
)  /\  { ( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) }  e.  ran  E ) ) )
8073, 79rspcedv 3211 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  (
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  /\  { ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 )
>. ) ) ,  ( lastS  `  W ) }  e.  ran  E )  ->  E. y  e.  ( ( V WWalksN  E
) `  N )
( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  /\  { ( lastS  `  y ) ,  ( lastS  `  W ) }  e.  ran  E ) ) )
811, 71, 80mp2and 679 . 2  |-  ( ( 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 ) )
8281ex 434 1  |-  ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   {cpr 4022   <.cop 4026   class class class wbr 4440   ran crn 4993   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    - cmin 9794   NNcn 10525   NN0cn0 10784   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   lastS clsw 12488   substr csubstr 12491   WWalksN cwwlkn 24340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-lsw 12496  df-substr 12499  df-wwlk 24341  df-wwlkn 24342
This theorem is referenced by:  wwlknredwwlkn0  24389
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