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Theorem wwlkextwrd 24593
Description: Lemma 0 for wwlkextbij 24598. (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 . 2  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E ) }
2 3anass 976 . . . . . 6  |-  ( ( ( # `  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
) ) )
32anbi2i 694 . . . . 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 ) ) ) )
4 anass 649 . . . . 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 ) ) ) )
53, 4bitr4i 252 . . . 4  |-  ( ( 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 ) ) )
6 wwlknprop 24551 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
7 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( 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 )
8 simpl 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  ->  w  e. Word  V )
98adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  e. Word  V
)
10 nn0re 10805 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  N  e.  RR )
11 2re 10606 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  e.  RR
1211a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  2  e.  RR )
13 nn0ge0 10822 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <_  N )
14 2pos 10628 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  <  2
1514a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <  2 )
1610, 12, 13, 15addgegt0d 10127 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
1716adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( N  +  2 ) )
18 breq2 4437 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
0  <  ( # `  w
)  <->  0  <  ( N  +  2 ) ) )
1918ad2antll 728 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( 0  < 
( # `  w )  <->  0  <  ( N  +  2 ) ) )
2017, 19mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( # `
 w ) )
21 hashgt0n0 12409 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  0  <  ( # `  w
) )  ->  w  =/=  (/) )
229, 20, 21syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  =/=  (/) )
23 lswcl 12563 . . . . . . . . . . . . . . . . 17  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  ( lastS  `  w )  e.  V
)
249, 22, 23syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( lastS  `  w )  e.  V )
2524adantrr 716 . . . . . . . . . . . . . . 15  |-  ( ( 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 )
26 swrdcl 12620 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( w  e. Word  V  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V )
27 eleq1 2513 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W  e. Word  V  <->  ( w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V ) )
2826, 27syl5ibr 221 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
2928eqcoms 2453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
3029adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  (
w  e. Word  V  ->  W  e. Word  V ) )
3130com12 31 . . . . . . . . . . . . . . . . . 18  |-  ( w  e. Word  V  ->  (
( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  W  e. Word  V ) )
3231adantr 465 . . . . . . . . . . . . . . . . 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 ) )
3332imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) )  ->  W  e. Word  V )
3433adantl 466 . . . . . . . . . . . . . . 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. Word  V )
35 oveq1 6284 . . . . . . . . . . . . . . . . . . 19  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
3635eqcoms 2453 . . . . . . . . . . . . . . . . . 18  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( W concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
3736adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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
) "> )
)
3837ad2antll 728 . . . . . . . . . . . . . . . 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 concat  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) concat  <" ( lastS  `  w ) "> ) )
39 oveq1 6284 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
4039adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( ( # `  w
)  -  1 )  =  ( ( N  +  2 )  - 
1 ) )
41 nn0cn 10806 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  N  e.  CC )
42 2cnd 10609 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  2  e.  CC )
43 1cnd 9610 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  1  e.  CC )
4441, 42, 43addsubassd 9951 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
45 2m1e1 10651 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 2  -  1 )  =  1
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
4746oveq2d 6293 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 ) )
4844, 47eqtrd 2482 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  1 ) )
4940, 48sylan9eqr 2504 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( # `  w )  -  1 )  =  ( N  +  1 ) )
5049opeq2d 4205 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  <. 0 ,  ( ( # `  w
)  -  1 )
>.  =  <. 0 ,  ( N  +  1 ) >. )
5150oveq2d 6293 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( w substr  <. 0 ,  ( ( # `  w )  -  1 ) >. )  =  ( w substr  <. 0 ,  ( N  +  1 )
>. ) )
5251oveq1d 6292 . . . . . . . . . . . . . . . . . 18  |-  ( ( 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
) "> )
)
53 swrdccatwrd 12667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  (
( w substr  <. 0 ,  ( ( # `  w
)  -  1 )
>. ) concat  <" ( lastS  `  w ) "> )  =  w )
549, 22, 53syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  w )
5552, 54eqtr3d 2484 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) concat  <" ( lastS  `  w
) "> )  =  w )
5655adantrr 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
) ) )  -> 
( ( w substr  <. 0 ,  ( N  + 
1 ) >. ) concat  <" ( lastS  `  w ) "> )  =  w )
5738, 56eqtr2d 2483 . . . . . . . . . . . . . . 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  =  ( W concat  <" ( lastS  `  w ) "> ) )
58 simprrr 764 . . . . . . . . . . . . . . 15  |-  ( ( 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 )
59 wwlknextbi 24590 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )
607, 25, 34, 57, 58, 59syl23anc 1234 . . . . . . . . . . . . . 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  +  1 ) )  <->  W  e.  ( ( V WWalksN  E
) `  N )
) )
6160exbiri 622 . . . . . . . . . . . . 13  |-  ( 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 ) ) ) ) )
6261com23 78 . . . . . . . . . . . 12  |-  ( 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 ) ) ) ) )
6362adantr 465 . . . . . . . . . . 11  |-  ( ( 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 ) ) ) ) )
6463adantl 466 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) ) )
656, 64mpcom 36 . . . . . . . . 9  |-  ( 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 ) ) ) )
6665expcomd 438 . . . . . . . 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 ) ) ) ) )
6766imp 429 . . . . . . 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 ) ) ) )
68 wwlknimp 24552 . . . . . . . . . . . 12  |-  ( 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
) )
6941, 43, 43addassd 9616 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
70 1p1e2 10650 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  +  1 )  =  2
7170a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( 1  +  1 )  =  2 )
7271oveq2d 6293 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( N  +  ( 1  +  1 ) )  =  ( N  +  2 ) )
7369, 72eqtrd 2482 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  2 ) )
7473eqeq2d 2455 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN0  ->  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  <->  ( # `  w
)  =  ( N  +  2 ) ) )
7574biimpd 207 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  ->  ( # `
 w )  =  ( N  +  2 ) ) )
7675adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( ( # `  w
)  =  ( ( N  +  1 )  +  1 )  -> 
( # `  w )  =  ( N  + 
2 ) ) )
7776com12 31 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  e.  NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
7877adantl 466 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
79 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  ->  w  e. Word  V )
8078, 79jctild 543 . . . . . . . . . . . . 13  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( w  e. Word  V  /\  ( # `  w )  =  ( N  +  2 ) ) ) )
81803adant3 1015 . . . . . . . . . . . 12  |-  ( ( 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 ) ) ) )
8268, 81syl 16 . . . . . . . . . . 11  |-  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) ) )
8382com12 31 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) ) )
8483adantl 466 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )
856, 84syl 16 . . . . . . . 8  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) ) ) )
8685adantr 465 . . . . . . 7  |-  ( ( 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 ) ) ) )
8767, 86impbid 191 . . . . . 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 ) ) ) )
8887ex 434 . . . . 5  |-  ( 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 ) ) ) ) )
8988pm5.32rd 640 . . . 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
) ) ) )
905, 89syl5bb 257 . . 3  |-  ( 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
) ) ) )
9190rabbidva2 3083 . 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 ) } )
921, 91syl5eq 2494 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   {crab 2795   _Vcvv 3093   (/)c0 3767   {cpr 4012   <.cop 4016   class class class wbr 4433   ran crn 4986   ` cfv 5574  (class class class)co 6277   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    < clt 9626    - cmin 9805   2c2 10586   NN0cn0 10796  ..^cfzo 11798   #chash 12379  Word cword 12508   lastS clsw 12509   concat cconcat 12510   <"cs1 12511   substr csubstr 12512   WWalksN cwwlkn 24543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-hash 12380  df-word 12516  df-lsw 12517  df-concat 12518  df-s1 12519  df-substr 12520  df-wwlk 24544  df-wwlkn 24545
This theorem is referenced by:  wwlkextsur  24596  wwlkextbij  24598
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