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Theorem wwlkextwrd 30285
Description: Lemma 0 for wwlkextbij 30290. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
Hypothesis
Ref Expression
wwlkextbij.d  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
Assertion
Ref Expression
wwlkextwrd  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  D  =  { w  e.  (
( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } )
Distinct variable groups:    w, E    w, N    w, V    w, W
Allowed substitution hint:    D( w)

Proof of Theorem wwlkextwrd
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wwlkextbij.d . . 3  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
21a1i 11 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  D  =  { w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )
3 3anass 964 . . . . . . 7  |-  ( ( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E )  <->  ( ( # `  w )  =  ( N  +  2 )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )
43anbi2i 689 . . . . . 6  |-  ( ( w  e. Word  V  /\  ( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  ( N  + 
2 )  /\  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) ) ) )
5 anass 644 . . . . . 6  |-  ( ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  <->  ( w  e. Word  V  /\  ( (
# `  w )  =  ( N  + 
2 )  /\  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) ) ) )
64, 5bitr4i 252 . . . . 5  |-  ( ( w  e. Word  V  /\  ( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  <->  ( (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  /\  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) ) )
7 wwlknprop 30245 . . . . . . . . . . 11  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
8 simpl 454 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  ->  N  e.  NN0 )
9 simpl 454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  ->  w  e. Word  V )
109adantl 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  e. Word  V
)
11 nn0re 10584 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
12 2re 10387 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  RR
1312a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  2  e.  RR )
14 nn0ge0 10601 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <_  N )
15 2pos 10409 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  <  2
1615a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <  2 )
1711, 13, 14, 16addgegt0d 9909 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
1817adantr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( N  +  2 ) )
19 breq2 4293 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
0  <  ( # `  w
)  <->  0  <  ( N  +  2 ) ) )
2019adantl 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( 0  <  ( # `
 w )  <->  0  <  ( N  +  2 ) ) )
2120adantl 463 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( 0  < 
( # `  w )  <->  0  <  ( N  +  2 ) ) )
2218, 21mpbird 232 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( # `
 w ) )
23 hashgt0n0 12129 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  0  <  ( # `  w
) )  ->  w  =/=  (/) )
2410, 22, 23syl2anc 656 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  =/=  (/) )
25 lswcl 12266 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  ( lastS  `  w )  e.  V
)
2610, 24, 25syl2anc 656 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( lastS  `  w )  e.  V )
2726adantrr 711 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  -> 
( lastS  `  w )  e.  V )
28 swrdcl 12311 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( w  e. Word  V  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V )
29 eleq1 2501 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W  e. Word  V  <->  ( w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V ) )
3028, 29syl5ibr 221 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
3130eqcoms 2444 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
3231adantr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  (
w  e. Word  V  ->  W  e. Word  V ) )
3332com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e. Word  V  ->  (
( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  W  e. Word  V ) )
3433adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
)  ->  W  e. Word  V ) )
3534imp 429 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  W  e. Word  V )
3635adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  ->  W  e. Word  V )
37 oveq1 6097 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
3837eqcoms 2444 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
3938adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  ( W concat  <" ( lastS  `  w
) "> )  =  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) concat  <" ( lastS  `  w
) "> )
)
4039adantl 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  ( W concat  <" ( lastS  `  w
) "> )  =  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) concat  <" ( lastS  `  w
) "> )
)
4140adantl 463 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  -> 
( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
42 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
4342adantl 463 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( ( # `  w
)  -  1 )  =  ( ( N  +  2 )  - 
1 ) )
44 nn0cn 10585 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  N  e.  CC )
45 2cnd 10390 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  2  e.  CC )
46 ax-1cn 9336 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  CC
4746a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  1  e.  CC )
4844, 45, 47addsubassd 9735 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
49 2m1e1 10432 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2  -  1 )  =  1
5049a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
5150oveq2d 6106 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 ) )
5248, 51eqtrd 2473 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  1 ) )
5343, 52sylan9eqr 2495 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( # `  w )  -  1 )  =  ( N  +  1 ) )
5453opeq2d 4063 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  <. 0 ,  ( ( # `  w
)  -  1 )
>.  =  <. 0 ,  ( N  +  1 ) >. )
5554oveq2d 6106 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( w substr  <. 0 ,  ( ( # `  w )  -  1 ) >. )  =  ( w substr  <. 0 ,  ( N  +  1 )
>. ) )
5655oveq1d 6105 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) concat  <" ( lastS  `  w
) "> )
)
57 swrdccatwrd 12358 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  (
( w substr  <. 0 ,  ( ( # `  w
)  -  1 )
>. ) concat  <" ( lastS  `  w ) "> )  =  w )
5810, 24, 57syl2anc 656 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  w )
5956, 58eqtr3d 2475 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  w )
6059adantrr 711 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  -> 
( ( w substr  <. 0 ,  ( N  + 
1 ) >. ) concat  <" ( lastS  `  w ) "> )  =  w )
6141, 60eqtr2d 2474 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  ->  w  =  ( W concat  <" ( lastS  `  w ) "> ) )
62 simprrr 759 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  ->  { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E )
63 wwlknextbi 30282 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  ( lastS  `  w )  e.  V )  /\  ( W  e. Word  V  /\  w  =  ( W concat  <" ( lastS  `  w ) "> )  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  ->  (
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  <->  W  e.  (
( V WWalksN  E ) `  N ) ) )
648, 27, 36, 61, 62, 63syl23anc 1220 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) )  -> 
( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  <->  W  e.  ( ( V WWalksN  E
) `  N )
) )
6564exbiri 619 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
6665com23 78 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
6766adantr 462 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( W  e.  ( ( V WWalksN  E ) `  N )  ->  (
( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
6867adantl 463 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) )  ->  ( W  e.  ( ( V WWalksN  E
) `  N )  ->  ( ( ( w  e. Word  V  /\  ( # `
 w )  =  ( N  +  2 ) )  /\  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  ->  w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
697, 68mpcom 36 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) )
7069exp3acom23 1420 . . . . . . . . 9  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E )  ->  ( (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  ->  w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) ) )
7170imp 429 . . . . . . . 8  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  ->  (
( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  ->  w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
72 wwlknimp 30246 . . . . . . . . . . . . 13  |-  ( 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
) )
7344, 47, 47addassd 9404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
74 1p1e2 10431 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 1  +  1 )  =  2
7574a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( 1  +  1 )  =  2 )
7675oveq2d 6106 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( N  +  ( 1  +  1 ) )  =  ( N  +  2 ) )
7773, 76eqtrd 2473 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  2 ) )
7877eqeq2d 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  <->  ( # `  w
)  =  ( N  +  2 ) ) )
7978biimpd 207 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN0  ->  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  ->  ( # `
 w )  =  ( N  +  2 ) ) )
8079adantr 462 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( ( # `  w
)  =  ( ( N  +  1 )  +  1 )  -> 
( # `  w )  =  ( N  + 
2 ) ) )
8180com12 31 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  e.  NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
8281adantl 463 . . . . . . . . . . . . . . 15  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
83 simpl 454 . . . . . . . . . . . . . . 15  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  ->  w  e. Word  V )
8482, 83jctild 540 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( w  e. Word  V  /\  ( # `  w )  =  ( N  +  2 ) ) ) )
85843adant3 1003 . . . . . . . . . . . . 13  |-  ( ( 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 )  ->  ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) ) ) )
8672, 85syl 16 . . . . . . . . . . . 12  |-  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) ) )
8786com12 31 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) ) )
8887adantl 463 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) )  ->  ( w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) ) ) )
897, 88syl 16 . . . . . . . . 9  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) ) ) )
9089adantr 462 . . . . . . . 8  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  ->  (
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( w  e. Word  V  /\  ( # `  w )  =  ( N  +  2 ) ) ) )
9171, 90impbid 191 . . . . . . 7  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  ->  (
( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  <-> 
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) ) ) )
9291ex 434 . . . . . 6  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E )  ->  ( (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  <->  w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) ) ) ) )
9392pm5.32rd 635 . . . . 5  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  <->  ( w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) ) )
946, 93syl5bb 257 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
w  e. Word  V  /\  ( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) )  <->  ( w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) ) ) )
9594abbidv 2555 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { w  |  ( w  e. Word  V  /\  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) ) }  =  { w  |  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) ) } )
96 df-rab 2722 . . 3  |-  { w  e. Word  V  |  ( (
# `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  =  {
w  |  ( w  e. Word  V  /\  (
( # `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) ) }
97 df-rab 2722 . . 3  |-  { w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  =  {
w  |  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) ) }
9895, 96, 973eqtr4g 2498 . 2  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  { w  e. Word  V  |  ( (
# `  w )  =  ( N  + 
2 )  /\  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) }  =  {
w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )
992, 98eqtrd 2473 1  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  D  =  { w  e.  (
( V WWalksN  E ) `  ( N  +  1 ) )  |  ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {cab 2427    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970   (/)c0 3634   {cpr 3876   <.cop 3880   class class class wbr 4289   ran crn 4837   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    - cmin 9591   2c2 10367   NN0cn0 10575  ..^cfzo 11544   #chash 12099  Word cword 12217   lastS clsw 12218   concat cconcat 12219   <"cs1 12220   substr csubstr 12221   WWalksN cwwlkn 30237
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-lsw 12226  df-concat 12227  df-s1 12228  df-substr 12229  df-wwlk 30238  df-wwlkn 30239
This theorem is referenced by:  wwlkextsur  30288  wwlkextbij  30290
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