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Theorem wwlkextwrd 24849
Description: Lemma 0 for wwlkextbij 24854. (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 975 . . . . . 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 692 . . . . 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 647 . . . . 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 24807 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
7 simpl 455 . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  ->  w  e. Word  V )
98adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  e. Word  V
)
10 nn0re 10721 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  N  e.  RR )
11 2re 10522 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  e.  RR
1211a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  2  e.  RR )
13 nn0ge0 10738 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <_  N )
14 2pos 10544 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  <  2
1514a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <  2 )
1610, 12, 13, 15addgegt0d 10043 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
1716adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  0  <  ( N  +  2 ) )
18 breq2 4371 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
0  <  ( # `  w
)  <->  0  <  ( N  +  2 ) ) )
1918ad2antll 726 . . . . . . . . . . . . . . . . . . 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 12338 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  0  <  ( # `  w
) )  ->  w  =/=  (/) )
229, 20, 21syl2anc 659 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  w  =/=  (/) )
23 lswcl 12497 . . . . . . . . . . . . . . . . 17  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  ( lastS  `  w )  e.  V
)
249, 22, 23syl2anc 659 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( lastS  `  w )  e.  V )
2524adantrr 714 . . . . . . . . . . . . . . 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 12555 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( w  e. Word  V  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  e. Word  V )
27 eleq1 2454 . . . . . . . . . . . . . . . . . . . . . 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 2394 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( w  e. Word  V  ->  W  e. Word  V ) )
3029adantr 463 . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . 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 6203 . . . . . . . . . . . . . . . . . . 19  |-  ( W  =  ( w substr  <. 0 ,  ( N  + 
1 ) >. )  ->  ( W ++  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) ++  <" ( lastS  `  w ) "> ) )
3635eqcoms 2394 . . . . . . . . . . . . . . . . . 18  |-  ( ( w substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( W ++  <" ( lastS  `  w ) "> )  =  ( (
w substr  <. 0 ,  ( N  +  1 )
>. ) ++  <" ( lastS  `  w ) "> ) )
3736adantr 463 . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 6203 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  w )  =  ( N  + 
2 )  ->  (
( # `  w )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
4039adantl 464 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) )  -> 
( ( # `  w
)  -  1 )  =  ( ( N  +  2 )  - 
1 ) )
41 nn0cn 10722 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  N  e.  CC )
42 2cnd 10525 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  2  e.  CC )
43 1cnd 9523 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  1  e.  CC )
4441, 42, 43addsubassd 9864 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
45 2m1e1 10567 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 2  -  1 )  =  1
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  ( 2  -  1 )  =  1 )
4746oveq2d 6212 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  ( 2  -  1 ) )  =  ( N  +  1 ) )
4844, 47eqtrd 2423 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  1 ) )
4940, 48sylan9eqr 2445 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( # `  w )  -  1 )  =  ( N  +  1 ) )
5049opeq2d 4138 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  <. 0 ,  ( ( # `  w
)  -  1 )
>.  =  <. 0 ,  ( N  +  1 ) >. )
5150oveq2d 6212 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( w substr  <. 0 ,  ( ( # `  w )  -  1 ) >. )  =  ( w substr  <. 0 ,  ( N  +  1 )
>. ) )
5251oveq1d 6211 . . . . . . . . . . . . . . . . . 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 12604 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  w  =/=  (/) )  ->  (
( w substr  <. 0 ,  ( ( # `  w
)  -  1 )
>. ) ++  <" ( lastS  `  w ) "> )  =  w )
549, 22, 53syl2anc 659 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( (
# `  w )  -  1 ) >.
) ++  <" ( lastS  `  w
) "> )  =  w )
5552, 54eqtr3d 2425 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  ( w  e. Word  V  /\  ( # `  w )  =  ( N  + 
2 ) ) )  ->  ( ( w substr  <. 0 ,  ( N  +  1 ) >.
) ++  <" ( lastS  `  w
) "> )  =  w )
5655adantrr 714 . . . . . . . . . . . . . . . 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 2424 . . . . . . . . . . . . . . 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 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 24846 . . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . . 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 620 . . . . . . . . . . . . 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 463 . . . . . . . . . . 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 464 . . . . . . . . . 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 436 . . . . . . . 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 427 . . . . . . 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 24808 . . . . . . . . . . . 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 9529 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  ( 1  +  1 ) ) )
70 1p1e2 10566 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  +  1 )  =  2
7170a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( 1  +  1 )  =  2 )
7271oveq2d 6212 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( N  +  ( 1  +  1 ) )  =  ( N  +  2 ) )
7369, 72eqtrd 2423 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  =  ( N  +  2 ) )
7473eqeq2d 2396 . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( # `  w
)  =  ( N  +  2 ) ) )
79 simpl 455 . . . . . . . . . . . . . 14  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  ->  w  e. Word  V )
8078, 79jctild 541 . . . . . . . . . . . . 13  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  ( ( N  +  1 )  +  1 ) )  -> 
( ( N  e. 
NN0  /\  W  e. Word  V )  ->  ( w  e. Word  V  /\  ( # `  w )  =  ( N  +  2 ) ) ) )
81803adant3 1014 . . . . . . . . . . . 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 464 . . . . . . . . 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 463 . . . . . . 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 432 . . . . 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 638 . . . 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 3024 . 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 2435 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   {crab 2736   _Vcvv 3034   (/)c0 3711   {cpr 3946   <.cop 3950   class class class wbr 4367   ran crn 4914   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    < clt 9539    - cmin 9718   2c2 10502   NN0cn0 10712  ..^cfzo 11717   #chash 12307  Word cword 12438   lastS clsw 12439   ++ cconcat 12440   <"cs1 12441   substr csubstr 12442   WWalksN cwwlkn 24799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-lsw 12447  df-concat 12448  df-s1 12449  df-substr 12450  df-wwlk 24800  df-wwlkn 24801
This theorem is referenced by:  wwlkextsur  24852  wwlkextbij  24854
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