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Theorem wwlkextwrd 25505
Description: Lemma 0 for wwlkextbij 25510. (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 995 . . . . . 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 705 . . . . 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 659 . . . . 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 260 . . . 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 25463 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
7 simpl 463 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  ->  w  e. Word  V )
98adantl 472 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  e. Word  V
)
10 nn0re 10907 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  N  e.  RR )
11 2re 10707 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  e.  RR
1211a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  2  e.  RR )
13 nn0ge0 10924 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <_  N )
14 2pos 10729 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  <  2
1514a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <  2 )
1610, 12, 13, 15addgegt0d 10215 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
1716adantr 471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( N  +  2 ) )
18 breq2 4420 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
0  <  ( # `  w
)  <->  0  <  ( N  +  2 ) ) )
1918ad2antll 740 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( 0  < 
( # `  w )  <->  0  <  ( N  +  2 ) ) )
2017, 19mpbird 240 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( # `
 w ) )
21 hashgt0n0 12578 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  0  <  ( # `  w
) )  ->  w  =/=  (/) )
229, 20, 21syl2anc 671 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  =/=  (/) )
23 lswcl 12751 . . . . . . . . . . . . . . . . 17  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  ( lastS  `  w )  e.  V
)
249, 22, 23syl2anc 671 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( lastS  `  w )  e.  V )
2524adantrr 728 . . . . . . . . . . . . . . 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 12812 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( w  e. Word  V  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V )
27 eleq1 2528 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W  e. Word  V  <->  ( w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V ) )
2826, 27syl5ibr 229 . . . . . . . . . . . . . . . . . . . . 21  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
2928eqcoms 2470 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
3029adantr 471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  (
w  e. Word  V  ->  W  e. Word  V ) )
3130com12 32 . . . . . . . . . . . . . . . . . 18  |-  ( w  e. Word  V  ->  (
( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  W  e. Word  V ) )
3231adantr 471 . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . . . . . . . . 19  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W ++  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) ++  <" ( lastS  `  w ) "> ) )
3635eqcoms 2470 . . . . . . . . . . . . . . . . . 18  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( W ++  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) ++  <" ( lastS  `  w ) "> ) )
3736adantr 471 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E )  ->  ( W ++  <" ( lastS  `  w
) "> )  =  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) ++  <" ( lastS  `  w
) "> )
)
3837ad2antll 740 . . . . . . . . . . . . . . . 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 ++  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) ++  <" ( lastS  `  w ) "> ) )
39 oveq1 6322 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
4039adantl 472 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( ( # `  w
)  -  1 )  =  ( ( N  +  2 )  - 
1 ) )
41 nn0cn 10908 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  N  e.  CC )
42 2cnd 10710 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  2  e.  CC )
43 1cnd 9685 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  1  e.  CC )
4441, 42, 43addsubassd 10032 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
45 2m1e1 10752 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 2  -  1 )  =  1
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
4746oveq2d 6331 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 ) )
4844, 47eqtrd 2496 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  1 ) )
4940, 48sylan9eqr 2518 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( # `  w )  -  1 )  =  ( N  +  1 ) )
5049opeq2d 4187 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  <. 0 ,  ( ( # `  w
)  -  1 )
>.  =  <. 0 ,  ( N  +  1 ) >. )
5150oveq2d 6331 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( w substr  <. 0 ,  ( ( # `  w )  -  1 ) >. )  =  ( w substr  <. 0 ,  ( N  +  1 )
>. ) )
5251oveq1d 6330 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) ++  <" ( lastS  `  w
) "> )  =  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) ++  <" ( lastS  `  w
) "> )
)
53 swrdccatwrd 12861 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  (
( w substr  <. 0 ,  ( ( # `  w
)  -  1 )
>. ) ++  <" ( lastS  `  w ) "> )  =  w )
549, 22, 53syl2anc 671 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) ++  <" ( lastS  `  w
) "> )  =  w )
5552, 54eqtr3d 2498 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) ++  <" ( lastS  `  w
) "> )  =  w )
5655adantrr 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 substr  <. 0 ,  ( N  + 
1 ) >. ) ++  <" ( lastS  `  w ) "> )  =  w )
5738, 56eqtr2d 2497 . . . . . . . . . . . . . . 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 ++  <" ( lastS  `  w ) "> ) )
58 simprrr 780 . . . . . . . . . . . . . . 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 25502 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN0  /\  ( lastS  `  w )  e.  V )  /\  ( W  e. Word  V  /\  w  =  ( W ++  <" ( 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 1283 . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . 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 81 . . . . . . . . . . . 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 471 . . . . . . . . . . 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 472 . . . . . . . . . 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 37 . . . . . . . . 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 444 . . . . . . . 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 435 . . . . . . 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 25464 . . . . . . . . . . . 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 9691 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
70 1p1e2 10751 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  +  1 )  =  2
7170a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( 1  +  1 )  =  2 )
7271oveq2d 6331 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( N  +  ( 1  +  1 ) )  =  ( N  +  2 ) )
7369, 72eqtrd 2496 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  2 ) )
7473eqeq2d 2472 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN0  ->  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  <->  ( # `  w
)  =  ( N  +  2 ) ) )
7574biimpd 212 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  NN0  ->  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  ->  ( # `
 w )  =  ( N  +  2 ) ) )
7675adantr 471 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( ( # `  w
)  =  ( ( N  +  1 )  +  1 )  -> 
( # `  w )  =  ( N  + 
2 ) ) )
7776com12 32 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  e.  NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
7877adantl 472 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
79 simpl 463 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  ->  w  e. Word  V )
8078, 79jctild 550 . . . . . . . . . . . . 13  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( w  e. Word  V  /\  ( # `  w )  =  ( N  +  2 ) ) ) )
81803adant3 1034 . . . . . . . . . . . 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 17 . . . . . . . . . . 11  |-  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) ) )
8382com12 32 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  W  e. Word  V )  ->  ( w  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  (
w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) ) )
8483adantl 472 . . . . . . . . 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 17 . . . . . . . 8  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( w  e.  ( ( V WWalksN  E
) `  ( N  +  1 ) )  ->  ( w  e. Word  V  /\  ( # `  w
)  =  ( N  +  2 ) ) ) )
8685adantr 471 . . . . . . 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 195 . . . . . 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 440 . . . . 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 650 . . . 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 265 . . 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 3046 . 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 2508 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   {crab 2753   _Vcvv 3057   (/)c0 3743   {cpr 3982   <.cop 3986   class class class wbr 4416   ran crn 4854   ` cfv 5601  (class class class)co 6315   RRcr 9564   0cc0 9565   1c1 9566    + caddc 9568    < clt 9701    - cmin 9886   2c2 10687   NN0cn0 10898  ..^cfzo 11946   #chash 12547  Word cword 12689   lastS clsw 12690   ++ cconcat 12691   <"cs1 12692   substr csubstr 12693   WWalksN cwwlkn 25455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-fzo 11947  df-hash 12548  df-word 12697  df-lsw 12698  df-concat 12699  df-s1 12700  df-substr 12701  df-wwlk 25456  df-wwlkn 25457
This theorem is referenced by:  wwlkextsur  25508  wwlkextbij  25510
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