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Theorem wwlkextinj 25028
Description: Lemma 2 for wwlkextbij 25031. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
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 ) }
wwlkextbij.r  |-  R  =  { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }
wwlkextbij.f  |-  F  =  ( t  e.  D  |->  ( lastS  `  t )
)
Assertion
Ref Expression
wwlkextinj  |-  ( N  e.  NN0  ->  F : D -1-1-> R )
Distinct variable groups:    t, D    n, E, w    t, N, w    t, R    n, V, t, w    n, W, t, w
Allowed substitution hints:    D( w, n)    R( w, n)    E( t)    F( w, t, n)    N( n)

Proof of Theorem wwlkextinj
Dummy variables  d  x are mutually distinct and 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 ) }
2 wwlkextbij.r . . 3  |-  R  =  { n  e.  V  |  { ( lastS  `  W
) ,  n }  e.  ran  E }
3 wwlkextbij.f . . 3  |-  F  =  ( t  e.  D  |->  ( lastS  `  t )
)
41, 2, 3wwlkextfun 25027 . 2  |-  ( N  e.  NN0  ->  F : D
--> R )
5 fveq2 5805 . . . . . . 7  |-  ( t  =  d  ->  ( lastS  `  t )  =  ( lastS  `  d ) )
6 fvex 5815 . . . . . . 7  |-  ( lastS  `  d
)  e.  _V
75, 3, 6fvmpt 5888 . . . . . 6  |-  ( d  e.  D  ->  ( F `  d )  =  ( lastS  `  d ) )
8 fveq2 5805 . . . . . . 7  |-  ( t  =  x  ->  ( lastS  `  t )  =  ( lastS  `  x ) )
9 fvex 5815 . . . . . . 7  |-  ( lastS  `  x
)  e.  _V
108, 3, 9fvmpt 5888 . . . . . 6  |-  ( x  e.  D  ->  ( F `  x )  =  ( lastS  `  x ) )
117, 10eqeqan12d 2425 . . . . 5  |-  ( ( d  e.  D  /\  x  e.  D )  ->  ( ( F `  d )  =  ( F `  x )  <-> 
( lastS  `  d )  =  ( lastS  `  x )
) )
1211adantl 464 . . . 4  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( F `  d
)  =  ( F `
 x )  <->  ( lastS  `  d
)  =  ( lastS  `  x
) ) )
13 fveq2 5805 . . . . . . . . 9  |-  ( w  =  d  ->  ( # `
 w )  =  ( # `  d
) )
1413eqeq1d 2404 . . . . . . . 8  |-  ( w  =  d  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  d
)  =  ( N  +  2 ) ) )
15 oveq1 6241 . . . . . . . . 9  |-  ( w  =  d  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
d substr  <. 0 ,  ( N  +  1 )
>. ) )
1615eqeq1d 2404 . . . . . . . 8  |-  ( w  =  d  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
17 fveq2 5805 . . . . . . . . . 10  |-  ( w  =  d  ->  ( lastS  `  w )  =  ( lastS  `  d ) )
1817preq2d 4057 . . . . . . . . 9  |-  ( w  =  d  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  d
) } )
1918eleq1d 2471 . . . . . . . 8  |-  ( w  =  d  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  d
) }  e.  ran  E ) )
2014, 16, 193anbi123d 1301 . . . . . . 7  |-  ( w  =  d  ->  (
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E )  <->  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) ) )
2120, 1elrab2 3208 . . . . . 6  |-  ( d  e.  D  <->  ( d  e. Word  V  /\  ( (
# `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) ) )
22 fveq2 5805 . . . . . . . . 9  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
2322eqeq1d 2404 . . . . . . . 8  |-  ( w  =  x  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  x
)  =  ( N  +  2 ) ) )
24 oveq1 6241 . . . . . . . . 9  |-  ( w  =  x  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. ) )
2524eqeq1d 2404 . . . . . . . 8  |-  ( w  =  x  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
26 fveq2 5805 . . . . . . . . . 10  |-  ( w  =  x  ->  ( lastS  `  w )  =  ( lastS  `  x ) )
2726preq2d 4057 . . . . . . . . 9  |-  ( w  =  x  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  x
) } )
2827eleq1d 2471 . . . . . . . 8  |-  ( w  =  x  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  x
) }  e.  ran  E ) )
2923, 25, 283anbi123d 1301 . . . . . . 7  |-  ( w  =  x  ->  (
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E )  <->  ( ( # `  x )  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  x ) }  e.  ran  E ) ) )
3029, 1elrab2 3208 . . . . . 6  |-  ( x  e.  D  <->  ( x  e. Word  V  /\  ( (
# `  x )  =  ( N  + 
2 )  /\  (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
31 eqtr3 2430 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( # `  x )  =  ( N  + 
2 ) )  -> 
( # `  d )  =  ( # `  x
) )
3231expcom 433 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( # `
 d )  =  ( # `  x
) ) )
33323ad2ant1 1018 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( ( # `
 d )  =  ( N  +  2 )  ->  ( # `  d
)  =  ( # `  x ) ) )
3433adantl 464 . . . . . . . . . . . . . 14  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( # `
 d )  =  ( # `  x
) ) )
3534com12 29 . . . . . . . . . . . . 13  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( # `
 d )  =  ( # `  x
) ) )
36353ad2ant1 1018 . . . . . . . . . . . 12  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E )  ->  ( (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( # `
 d )  =  ( # `  x
) ) )
3736adantl 464 . . . . . . . . . . 11  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  ->  (
( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( # `
 d )  =  ( # `  x
) ) )
3837imp 427 . . . . . . . . . 10  |-  ( ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  -> 
( # `  d )  =  ( # `  x
) )
3938adantr 463 . . . . . . . . 9  |-  ( ( ( ( d  e. Word  V  /\  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  -> 
( # `  d )  =  ( # `  x
) )
4039adantr 463 . . . . . . . 8  |-  ( ( ( ( ( d  e. Word  V  /\  (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  ( # `  d
)  =  ( # `  x ) )
41 simpr 459 . . . . . . . 8  |-  ( ( ( ( ( d  e. Word  V  /\  (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  ( lastS  `  d
)  =  ( lastS  `  x
) )
42 eqtr3 2430 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( d substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W
)  ->  ( d substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
) )
43 1e2m1 10612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  1  =  ( 2  -  1 )
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  1  =  ( 2  -  1 ) )
4544oveq2d 6250 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( N  +  ( 2  -  1 ) ) )
46 nn0cn 10766 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  N  e.  CC )
47 2cnd 10569 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  2  e.  CC )
48 1cnd 9562 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  1  e.  CC )
4946, 47, 48addsubassd 9907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
5045, 49eqtr4d 2446 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( ( N  + 
2 )  -  1 ) )
5150adantr 463 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( N  +  1 )  =  ( ( N  +  2 )  -  1 ) )
52 oveq1 6241 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( # `  d )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
5352eqeq2d 2416 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( N  +  1 )  =  ( (
# `  d )  -  1 )  <->  ( N  +  1 )  =  ( ( N  + 
2 )  -  1 ) ) )
5453adantl 464 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( ( N  + 
1 )  =  ( ( # `  d
)  -  1 )  <-> 
( N  +  1 )  =  ( ( N  +  2 )  -  1 ) ) )
5551, 54mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( N  +  1 )  =  ( (
# `  d )  -  1 ) )
56 opeq2 4159 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )
5756oveq2d 6250 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( d substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( d substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
5856oveq2d 6250 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
5957, 58eqeq12d 2424 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. )  <->  ( d substr  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d )  -  1 ) >. ) ) )
6055, 59syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
)  <->  ( d substr  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d )  -  1 ) >. ) ) )
6160biimpd 207 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
)  ->  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) )
6261ex 432 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  ( x substr  <. 0 ,  ( N  +  1 )
>. )  ->  ( d substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
6362com13 80 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. )  ->  ( (
# `  d )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
6442, 63syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( d substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W
)  ->  ( ( # `
 d )  =  ( N  +  2 )  ->  ( N  e.  NN0  ->  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
6564ex 432 . . . . . . . . . . . . . . . . . 18  |-  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  ->  ( (
# `  d )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) ) )
6665com23 78 . . . . . . . . . . . . . . . . 17  |-  ( ( d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( ( # `  d
)  =  ( N  +  2 )  -> 
( ( x substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  ->  ( N  e.  NN0  ->  ( d substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) ) )
6766impcom 428 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W )  ->  ( (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( N  e.  NN0  ->  ( d substr  <. 0 ,  ( ( # `  d )  -  1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
6867com12 29 . . . . . . . . . . . . . . 15  |-  ( ( x substr  <. 0 ,  ( N  +  1 )
>. )  =  W  ->  ( ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W )  ->  ( N  e. 
NN0  ->  ( d substr  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d )  -  1 ) >. ) ) ) )
69683ad2ant2 1019 . . . . . . . . . . . . . 14  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W
)  ->  ( N  e.  NN0  ->  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
7069adantl 464 . . . . . . . . . . . . 13  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  (
( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W )  ->  ( N  e.  NN0  ->  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
7170com12 29 . . . . . . . . . . . 12  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W )  ->  ( (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
72713adant3 1017 . . . . . . . . . . 11  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E )  ->  ( (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
7372adantl 464 . . . . . . . . . 10  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  ->  (
( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
7473imp31 430 . . . . . . . . 9  |-  ( ( ( ( d  e. Word  V  /\  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  -> 
( d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) )
7574adantr 463 . . . . . . . 8  |-  ( ( ( ( ( d  e. Word  V  /\  (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
76 simpl 455 . . . . . . . . . . . . 13  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  ->  d  e. Word  V )
77 simpl 455 . . . . . . . . . . . . 13  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  x  e. Word  V )
7876, 77anim12i 564 . . . . . . . . . . . 12  |-  ( ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  -> 
( d  e. Word  V  /\  x  e. Word  V ) )
7978adantr 463 . . . . . . . . . . 11  |-  ( ( ( ( d  e. Word  V  /\  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  -> 
( d  e. Word  V  /\  x  e. Word  V ) )
80 nn0re 10765 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  N  e.  RR )
81 2re 10566 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  e.  RR
8281a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  2  e.  RR )
83 nn0ge0 10782 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <_  N )
84 2pos 10588 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  <  2
8584a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <  2 )
8680, 82, 83, 85addgegt0d 10086 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
8786adantl 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
88 breq2 4398 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
0  <  ( # `  d
)  <->  0  <  ( N  +  2 ) ) )
8988adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
( 0  <  ( # `
 d )  <->  0  <  ( N  +  2 ) ) )
9087, 89mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( # `  d
) )
91 hashgt0n0 12390 . . . . . . . . . . . . . . . . . 18  |-  ( ( d  e. Word  V  /\  0  <  ( # `  d
) )  ->  d  =/=  (/) )
9290, 91sylan2 472 . . . . . . . . . . . . . . . . 17  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 ) )  ->  d  =/=  (/) )
9392exp32 603 . . . . . . . . . . . . . . . 16  |-  ( d  e. Word  V  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
9493com12 29 . . . . . . . . . . . . . . 15  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
d  e. Word  V  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
95943ad2ant1 1018 . . . . . . . . . . . . . 14  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E )  ->  ( d  e. Word  V  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
9695impcom 428 . . . . . . . . . . . . 13  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  d  =/=  (/) ) )
9796adantr 463 . . . . . . . . . . . 12  |-  ( ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  -> 
( N  e.  NN0  ->  d  =/=  (/) ) )
9897imp 427 . . . . . . . . . . 11  |-  ( ( ( ( d  e. Word  V  /\  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  -> 
d  =/=  (/) )
9986adantl 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
100 breq2 4398 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
0  <  ( # `  x
)  <->  0  <  ( N  +  2 ) ) )
101100adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
( 0  <  ( # `
 x )  <->  0  <  ( N  +  2 ) ) )
10299, 101mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( # `  x
) )
103 hashgt0n0 12390 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e. Word  V  /\  0  <  ( # `  x
) )  ->  x  =/=  (/) )
104102, 103sylan2 472 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 ) )  ->  x  =/=  (/) )
105104exp32 603 . . . . . . . . . . . . . . . 16  |-  ( x  e. Word  V  ->  (
( # `  x )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
106105com12 29 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
x  e. Word  V  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
1071063ad2ant1 1018 . . . . . . . . . . . . . 14  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( x  e. Word  V  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
108107impcom 428 . . . . . . . . . . . . 13  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  x  =/=  (/) ) )
109108adantl 464 . . . . . . . . . . . 12  |-  ( ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  -> 
( N  e.  NN0  ->  x  =/=  (/) ) )
110109imp 427 . . . . . . . . . . 11  |-  ( ( ( ( d  e. Word  V  /\  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  ->  x  =/=  (/) )
11179, 98, 110jca32 533 . . . . . . . . . 10  |-  ( ( ( ( d  e. Word  V  /\  ( ( # `  d )  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  d ) }  e.  ran  E
) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  -> 
( ( d  e. Word  V  /\  x  e. Word  V
)  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) ) )
112111adantr 463 . . . . . . . . 9  |-  ( ( ( ( ( d  e. Word  V  /\  (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  ( (
d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) ) )
113 simpl 455 . . . . . . . . . . . 12  |-  ( ( d  e. Word  V  /\  x  e. Word  V )  ->  d  e. Word  V )
114113adantr 463 . . . . . . . . . . 11  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  d  e. Word  V
)
115 simpr 459 . . . . . . . . . . . 12  |-  ( ( d  e. Word  V  /\  x  e. Word  V )  ->  x  e. Word  V )
116115adantr 463 . . . . . . . . . . 11  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  x  e. Word  V
)
117 hashneq0 12389 . . . . . . . . . . . . . . . 16  |-  ( d  e. Word  V  ->  (
0  <  ( # `  d
)  <->  d  =/=  (/) ) )
118117biimprd 223 . . . . . . . . . . . . . . 15  |-  ( d  e. Word  V  ->  (
d  =/=  (/)  ->  0  <  ( # `  d
) ) )
119118adantr 463 . . . . . . . . . . . . . 14  |-  ( ( d  e. Word  V  /\  x  e. Word  V )  ->  ( d  =/=  (/)  ->  0  <  ( # `  d
) ) )
120119com12 29 . . . . . . . . . . . . 13  |-  ( d  =/=  (/)  ->  ( (
d  e. Word  V  /\  x  e. Word  V )  ->  0  <  ( # `  d ) ) )
121120adantr 463 . . . . . . . . . . . 12  |-  ( ( d  =/=  (/)  /\  x  =/=  (/) )  ->  (
( d  e. Word  V  /\  x  e. Word  V )  ->  0  <  ( # `
 d ) ) )
122121impcom 428 . . . . . . . . . . 11  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  0  <  ( # `
 d ) )
123 2swrd1eqwrdeq 12642 . . . . . . . . . . 11  |-  ( ( d  e. Word  V  /\  x  e. Word  V  /\  0  <  ( # `  d
) )  ->  (
d  =  x  <->  ( ( # `
 d )  =  ( # `  x
)  /\  ( (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  /\  ( lastS  `  d )  =  ( lastS  `  x ) ) ) ) )
124114, 116, 122, 123syl3anc 1230 . . . . . . . . . 10  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  ( d  =  x  <->  ( ( # `  d )  =  (
# `  x )  /\  ( ( d substr  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d )  -  1 ) >. )  /\  ( lastS  `  d )  =  ( lastS  `  x ) ) ) ) )
125 ancom 448 . . . . . . . . . . . 12  |-  ( ( ( d substr  <. 0 ,  ( ( # `  d )  -  1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  /\  ( lastS  `  d )  =  ( lastS  `  x ) )  <->  ( ( lastS  `  d )  =  ( lastS  `  x )  /\  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) )
126125anbi2i 692 . . . . . . . . . . 11  |-  ( ( ( # `  d
)  =  ( # `  x )  /\  (
( d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  /\  ( lastS  `  d )  =  ( lastS  `  x ) ) )  <-> 
( ( # `  d
)  =  ( # `  x )  /\  (
( lastS  `  d )  =  ( lastS  `  x )  /\  ( d substr  <. 0 ,  ( ( # `  d )  -  1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
127 3anass 978 . . . . . . . . . . 11  |-  ( ( ( # `  d
)  =  ( # `  x )  /\  ( lastS  `  d )  =  ( lastS  `  x )  /\  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) )  <->  ( ( # `
 d )  =  ( # `  x
)  /\  ( ( lastS  `  d )  =  ( lastS  `  x )  /\  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
128126, 127bitr4i 252 . . . . . . . . . 10  |-  ( ( ( # `  d
)  =  ( # `  x )  /\  (
( d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  /\  ( lastS  `  d )  =  ( lastS  `  x ) ) )  <-> 
( ( # `  d
)  =  ( # `  x )  /\  ( lastS  `  d )  =  ( lastS  `  x )  /\  (
d substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) )
129124, 128syl6bb 261 . . . . . . . . 9  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  ( d  =  x  <->  ( ( # `  d )  =  (
# `  x )  /\  ( lastS  `  d )  =  ( lastS  `  x
)  /\  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
130112, 129syl 17 . . . . . . . 8  |-  ( ( ( ( ( d  e. Word  V  /\  (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  ( d  =  x  <->  ( ( # `  d )  =  (
# `  x )  /\  ( lastS  `  d )  =  ( lastS  `  x
)  /\  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
13140, 41, 75, 130mpbir3and 1180 . . . . . . 7  |-  ( ( ( ( ( d  e. Word  V  /\  (
( # `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  /\  N  e.  NN0 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  d  =  x )
132131exp31 602 . . . . . 6  |-  ( ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  /\  (
x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )  -> 
( N  e.  NN0  ->  ( ( lastS  `  d
)  =  ( lastS  `  x
)  ->  d  =  x ) ) )
13321, 30, 132syl2anb 477 . . . . 5  |-  ( ( d  e.  D  /\  x  e.  D )  ->  ( N  e.  NN0  ->  ( ( lastS  `  d
)  =  ( lastS  `  x
)  ->  d  =  x ) ) )
134133impcom 428 . . . 4  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( lastS  `  d )  =  ( lastS  `  x )  ->  d  =  x ) )
13512, 134sylbid 215 . . 3  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( F `  d
)  =  ( F `
 x )  -> 
d  =  x ) )
136135ralrimivva 2824 . 2  |-  ( N  e.  NN0  ->  A. d  e.  D  A. x  e.  D  ( ( F `  d )  =  ( F `  x )  ->  d  =  x ) )
137 dff13 6103 . 2  |-  ( F : D -1-1-> R  <->  ( F : D --> R  /\  A. d  e.  D  A. x  e.  D  (
( F `  d
)  =  ( F `
 x )  -> 
d  =  x ) ) )
1384, 136, 137sylanbrc 662 1  |-  ( N  e.  NN0  ->  F : D -1-1-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   {crab 2757   (/)c0 3737   {cpr 3973   <.cop 3977   class class class wbr 4394    |-> cmpt 4452   ran crn 4943   -->wf 5521   -1-1->wf1 5522   ` cfv 5525  (class class class)co 6234   RRcr 9441   0cc0 9442   1c1 9443    + caddc 9445    < clt 9578    - cmin 9761   2c2 10546   NN0cn0 10756   #chash 12359  Word cword 12490   lastS clsw 12491   substr csubstr 12494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-fzo 11768  df-hash 12360  df-word 12498  df-lsw 12499  df-s1 12501  df-substr 12502
This theorem is referenced by:  wwlkextbij0  25030
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