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Theorem wwlkextwrd 30365
Description: Lemma 0 for wwlkextbij 30370. (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 969 . . . . . . 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 694 . . . . . 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 649 . . . . . 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 30325 . . . . . . . . . . 11  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
8 simpl 457 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  ->  w  e. Word  V )
109adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  e. Word  V
)
11 nn0re 10593 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
12 2re 10396 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  RR
1312a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  2  e.  RR )
14 nn0ge0 10610 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <_  N )
15 2pos 10418 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  <  2
1615a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <  2 )
1711, 13, 14, 16addgegt0d 9918 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
1817adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( N  +  2 ) )
19 breq2 4301 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
0  <  ( # `  w
)  <->  0  <  ( N  +  2 ) ) )
2019adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( 0  <  ( # `
 w )  <->  0  <  ( N  +  2 ) ) )
2120adantl 466 . . . . . . . . . . . . . . . . . . . 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 12138 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  0  <  ( # `  w
) )  ->  w  =/=  (/) )
2410, 22, 23syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  =/=  (/) )
25 lswcl 12275 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  ( lastS  `  w )  e.  V
)
2610, 24, 25syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( lastS  `  w )  e.  V )
2726adantrr 716 . . . . . . . . . . . . . . . 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 12320 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( w  e. Word  V  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V )
29 eleq1 2503 . . . . . . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
3231adantr 465 . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
3837eqcoms 2446 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
3938adantr 465 . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
4342adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( ( # `  w
)  -  1 )  =  ( ( N  +  2 )  - 
1 ) )
44 nn0cn 10594 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  N  e.  CC )
45 2cnd 10399 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  2  e.  CC )
46 ax-1cn 9345 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  CC
4746a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  1  e.  CC )
4844, 45, 47addsubassd 9744 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
49 2m1e1 10441 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2  -  1 )  =  1
5049a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
5150oveq2d 6112 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 ) )
5248, 51eqtrd 2475 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  1 ) )
5343, 52sylan9eqr 2497 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( # `  w )  -  1 )  =  ( N  +  1 ) )
5453opeq2d 4071 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  <. 0 ,  ( ( # `  w
)  -  1 )
>.  =  <. 0 ,  ( N  +  1 ) >. )
5554oveq2d 6112 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( w substr  <. 0 ,  ( ( # `  w )  -  1 ) >. )  =  ( w substr  <. 0 ,  ( N  +  1 )
>. ) )
5655oveq1d 6111 . . . . . . . . . . . . . . . . . . 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 12367 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  (
( w substr  <. 0 ,  ( ( # `  w
)  -  1 )
>. ) concat  <" ( lastS  `  w ) "> )  =  w )
5810, 24, 57syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  w )
5956, 58eqtr3d 2477 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  w )
6059adantrr 716 . . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . 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 30362 . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . 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 622 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 ) ) ) )
7069expcomd 438 . . . . . . . . 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 30326 . . . . . . . . . . . . 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 9413 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
74 1p1e2 10440 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 1  +  1 )  =  2
7574a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( 1  +  1 )  =  2 )
7675oveq2d 6112 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( N  +  ( 1  +  1 ) )  =  ( N  +  2 ) )
7773, 76eqtrd 2475 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  2 ) )
7877eqeq2d 2454 . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . 15  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
83 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  ->  w  e. Word  V )
8482, 83jctild 543 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( w  e. Word  V  /\  ( # `  w )  =  ( N  +  2 ) ) ) )
85843adant3 1008 . . . . . . . . . . . . 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 466 . . . . . . . . . 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 465 . . . . . . . 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 640 . . . . 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 2562 . . 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 2729 . . 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 2729 . . 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 2500 . 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 2475 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 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977   (/)c0 3642   {cpr 3884   <.cop 3888   class class class wbr 4297   ran crn 4846   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    < clt 9423    - cmin 9600   2c2 10376   NN0cn0 10584  ..^cfzo 11553   #chash 12108  Word cword 12226   lastS clsw 12227   concat cconcat 12228   <"cs1 12229   substr csubstr 12230   WWalksN cwwlkn 30317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-lsw 12235  df-concat 12236  df-s1 12237  df-substr 12238  df-wwlk 30318  df-wwlkn 30319
This theorem is referenced by:  wwlkextsur  30368  wwlkextbij  30370
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