MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlknredwwlkn Structured version   Unicode version

Theorem wwlknredwwlkn 25143
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 2403 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  =  ( W substr  <. 0 ,  ( N  +  1 )
>. ) )
2 wwlknimp 25104 . . . 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 756 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  W  e. Word  V
)
4 peano2nn0 10877 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5 peano2nn0 10877 . . . . . . . . . . . . 13  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e. 
NN0 )
64, 5syl 17 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e. 
NN0 )
7 id 22 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e. 
NN0 )
8 nn0p1nn 10876 . . . . . . . . . . . . . 14  |-  ( ( N  +  1 )  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  NN )
94, 8syl 17 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  NN )
10 nn0re 10845 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  N  e.  RR )
11 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  N  e.  RR )
12 peano2re 9787 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
13 peano2re 9787 . . . . . . . . . . . . . . . . 17  |-  ( ( N  +  1 )  e.  RR  ->  (
( N  +  1 )  +  1 )  e.  RR )
1412, 13syl 17 . . . . . . . . . . . . . . . 16  |-  ( N  e.  RR  ->  (
( N  +  1 )  +  1 )  e.  RR )
1511, 12, 143jca 1177 . . . . . . . . . . . . . . 15  |-  ( N  e.  RR  ->  ( N  e.  RR  /\  ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  +  1 )  e.  RR ) )
1610, 15syl 17 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( N  e.  RR  /\  ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  +  1 )  e.  RR ) )
1710ltp1d 10516 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  N  < 
( N  +  1 ) )
18 nn0re 10845 . . . . . . . . . . . . . . . 16  |-  ( ( N  +  1 )  e.  NN0  ->  ( N  +  1 )  e.  RR )
194, 18syl 17 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  RR )
2019ltp1d 10516 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( N  +  1 )  < 
( ( N  + 
1 )  +  1 ) )
21 lttr 9692 . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  < 
( ( N  + 
1 )  +  1 ) )
24 elfzo0 11895 . . . . . . . . . . . . 13  |-  ( N  e.  ( 0..^ ( ( N  +  1 )  +  1 ) )  <->  ( N  e. 
NN0  /\  ( ( N  +  1 )  +  1 )  e.  NN  /\  N  < 
( ( N  + 
1 )  +  1 ) ) )
257, 9, 23, 24syl3anbrc 1181 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( ( N  +  1 )  +  1 ) ) )
26 fargshiftlem 25051 . . . . . . . . . . . 12  |-  ( ( ( ( N  + 
1 )  +  1 )  e.  NN0  /\  N  e.  ( 0..^ ( ( N  + 
1 )  +  1 ) ) )  -> 
( N  +  1 )  e.  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
276, 25, 26syl2anc 659 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) )
2827adantr 463 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  ( N  + 
1 )  e.  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
29 oveq2 6286 . . . . . . . . . . . . 13  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
1 ... ( # `  W
) )  =  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
3029eleq2d 2472 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  +  1 )  e.  ( 1 ... ( # `  W
) )  <->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) ) )
3130adantl 464 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  + 
1 )  e.  ( 1 ... ( # `  W ) )  <->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) ) )
3231adantl 464 . . . . . . . . . 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 530 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) ) )  ->  ( W  e. Word  V  /\  ( N  + 
1 )  e.  ( 1 ... ( # `  W ) ) ) )
35343adantr3 1158 . . . . . . 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 12724 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( N  +  1
)  e.  ( 1 ... ( # `  W
) ) )  -> 
( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
)  =  ( W `
 ( ( N  +  1 )  - 
1 ) ) )
3735, 36syl 17 . . . . . 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 12638 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
39383ad2ant1 1018 . . . . . . 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 464 . . . . . 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 4059 . . . . 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 6285 . . . . . . . . . . 11  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( # `  W )  -  1 )  =  ( ( ( N  +  1 )  +  1 )  -  1 ) )
43423ad2ant2 1019 . . . . . . . . . 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 464 . . . . . . . . 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 5853 . . . . . . . 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 4058 . . . . . . 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 10846 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  CC )
48 1cnd 9642 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  e.  CC )
4947, 48pncand 9968 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  -  1 )  =  N )
5049fveq2d 5853 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 ( ( N  +  1 )  - 
1 ) )  =  ( W `  N
) )
514nn0cnd 10895 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  CC )
5251, 48pncand 9968 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( ( ( N  +  1 )  +  1 )  -  1 )  =  ( N  +  1 ) )
5352fveq2d 5853 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( W `
 ( ( ( N  +  1 )  +  1 )  - 
1 ) )  =  ( W `  ( N  +  1 ) ) )
5450, 53preq12d 4059 . . . . . . . 8  |-  ( N  e.  NN0  ->  { ( W `  ( ( N  +  1 )  -  1 ) ) ,  ( W `  ( ( ( N  +  1 )  +  1 )  -  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1 ) ) } )
5554adantr 463 . . . . . . 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
) ) } )
5646, 55eqtrd 2443 . . . . . 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
) ) } )
57 fzonn0p1 11928 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ( 0..^ ( N  +  1 ) ) )
58 fveq2 5849 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  ( W `  i )  =  ( W `  N ) )
59 oveq1 6285 . . . . . . . . . . . . . 14  |-  ( i  =  N  ->  (
i  +  1 )  =  ( N  + 
1 ) )
6059fveq2d 5853 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  ( W `  ( i  +  1 ) )  =  ( W `  ( N  +  1
) ) )
6158, 60preq12d 4059 . . . . . . . . . . . 12  |-  ( i  =  N  ->  { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  =  { ( W `  N ) ,  ( W `  ( N  +  1 ) ) } )
6261eleq1d 2471 . . . . . . . . . . 11  |-  ( i  =  N  ->  ( { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( W `  N
) ,  ( W `
 ( N  + 
1 ) ) }  e.  ran  E ) )
6362rspcv 3156 . . . . . . . . . 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 ) )
6457, 63syl 17 . . . . . . . . 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 ) )
6564com12 29 . . . . . . . 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 ) )
66653ad2ant3 1020 . . . . . . 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 ) )
6766impcom 428 . . . . . 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
)
6856, 67eqeltrd 2490 . . . . 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
)
6941, 68eqeltrd 2490 . . . 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 )
702, 69sylan2 472 . . 3  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  { ( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) }  e.  ran  E )
71 wwlknred 25140 . . . . 5  |-  ( N  e.  NN0  ->  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W substr  <.
0 ,  ( N  +  1 ) >.
)  e.  ( ( V WWalksN  E ) `  N
) ) )
7271imp 427 . . . 4  |-  ( ( N  e.  NN0  /\  W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) )  ->  ( W substr  <. 0 ,  ( N  +  1 )
>. )  e.  (
( V WWalksN  E ) `  N ) )
73 eqeq2 2417 . . . . . 6  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  y  <->  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) )
74 fveq2 5849 . . . . . . . 8  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( lastS  `  y )  =  ( lastS  `  ( W substr  <. 0 ,  ( N  +  1 ) >.
) ) )
7574preq1d 4057 . . . . . . 7  |-  ( y  =  ( W substr  <. 0 ,  ( N  + 
1 ) >. )  ->  { ( lastS  `  y
) ,  ( lastS  `  W
) }  =  {
( lastS  `  ( W substr  <. 0 ,  ( N  + 
1 ) >. )
) ,  ( lastS  `  W
) } )
7675eleq1d 2471 . . . . . 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 ) )
7773, 76anbi12d 709 . . . . 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 ) ) )
7877adantl 464 . . . 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 ) ) )
7972, 78rspcedv 3164 . . 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 ) ) )
801, 70, 79mp2and 677 . 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 ) )
8180ex 432 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   {cpr 3974   <.cop 3978   class class class wbr 4395   ran crn 4824   ` cfv 5569  (class class class)co 6278   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    < clt 9658    - cmin 9841   NNcn 10576   NN0cn0 10836   ...cfz 11726  ..^cfzo 11854   #chash 12452  Word cword 12583   lastS clsw 12584   substr csubstr 12587   WWalksN cwwlkn 25095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-lsw 12592  df-substr 12595  df-wwlk 25096  df-wwlkn 25097
This theorem is referenced by:  wwlknredwwlkn0  25144
  Copyright terms: Public domain W3C validator