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Theorem wwlknredwwlkn 30496
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 2452 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  ( W substr  <. 0 ,  ( N  +  1 )
>. ) )
2 wwlknimp 30459 . . . 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 10721 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5 peano2nn0 10721 . . . . . . . . . . . . 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 10720 . . . . . . . . . . . . . 14  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  NN )
94, 8syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  NN )
10 nn0re 10689 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  N  e.  RR )
11 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  N  e.  RR )
12 peano2re 9643 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
13 peano2re 9643 . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . 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 10364 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  < 
( N  +  1 ) )
18 nn0re 10689 . . . . . . . . . . . . . . . 16  |-  ( ( N  +  1 )  e.  NN0  ->  ( N  +  1 )  e.  RR )
194, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  RR )
2019ltp1d 10364 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( N  +  1 )  < 
( ( N  + 
1 )  +  1 ) )
21 lttr 9552 . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  < 
( ( N  + 
1 )  +  1 ) )
24 elfzo0 11688 . . . . . . . . . . . . 13  |-  ( N  e.  ( 0..^ ( ( N  +  1 )  +  1 ) )  <->  ( N  e. 
NN0  /\  ( ( N  +  1 )  +  1 )  e.  NN  /\  N  < 
( ( N  + 
1 )  +  1 ) ) )
257, 9, 23, 24syl3anbrc 1172 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( ( N  +  1 )  +  1 ) ) )
26 fargshiftlem 23655 . . . . . . . . . . . 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 6198 . . . . . . . . . . . . 13  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
1 ... ( # `  W
) )  =  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
3029eleq2d 2521 . . . . . . . . . . . 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 1149 . . . . . . 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 12441 . . . . . . 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 12368 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
39383ad2ant1 1009 . . . . . . 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 4060 . . . . 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 6197 . . . . . . . . . . 11  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( # `  W )  -  1 )  =  ( ( ( N  +  1 )  +  1 )  -  1 ) )
43423ad2ant2 1010 . . . . . . . . . 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 5793 . . . . . . . 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 4059 . . . . . . 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 10690 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
48 ax-1cn 9441 . . . . . . . . . . . 12  |-  1  e.  CC
4948a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  e.  CC )
5047, 49pncand 9821 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  -  1 )  =  N )
5150fveq2d 5793 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 ( ( N  +  1 )  - 
1 ) )  =  ( W `  N
) )
524nn0cnd 10739 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  CC )
5352, 49pncand 9821 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( ( ( N  +  1 )  +  1 )  -  1 )  =  ( N  +  1 ) )
5453fveq2d 5793 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 ( ( ( N  +  1 )  +  1 )  - 
1 ) )  =  ( W `  ( N  +  1 ) ) )
5551, 54preq12d 4060 . . . . . . . 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 2492 . . . . . 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 11710 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( N  +  1 ) ) )
59 fveq2 5789 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  ( W `  i )  =  ( W `  N ) )
60 oveq1 6197 . . . . . . . . . . . . . 14  |-  ( i  =  N  ->  (
i  +  1 )  =  ( N  + 
1 ) )
6160fveq2d 5793 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  ( W `  ( i  +  1 ) )  =  ( W `  ( N  +  1
) ) )
6259, 61preq12d 4060 . . . . . . . . . . . 12  |-  ( i  =  N  ->  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1 ) ) } )
6362eleq1d 2520 . . . . . . . . . . 11  |-  ( i  =  N  ->  ( { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  N
) ,  ( W `
 ( N  + 
1 ) ) }  e.  ran  E ) )
6463rspcv 3165 . . . . . . . . . 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 1011 . . . . . . 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 2539 . . . . 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 2539 . . . 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 30493 . . . . 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 2466 . . . . . 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 5789 . . . . . . . 8  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( lastS  `  y )  =  ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 ) >.
) ) )
7675preq1d 4058 . . . . . . 7  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  { ( lastS  `  y
) ,  ( lastS  `  W
) }  =  {
( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) } )
7776eleq1d 2520 . . . . . 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 3173 . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {cpr 3977   <.cop 3981   class class class wbr 4390   ran crn 4939   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    + caddc 9386    < clt 9519    - cmin 9696   NNcn 10423   NN0cn0 10680   ...cfz 11538  ..^cfzo 11649   #chash 12204  Word cword 12323   lastS clsw 12324   substr csubstr 12327   WWalksN cwwlkn 30450
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-hash 12205  df-word 12331  df-lsw 12332  df-substr 12335  df-wwlk 30451  df-wwlkn 30452
This theorem is referenced by:  wwlknredwwlkn0  30497
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