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Theorem wwlkextinj 25470
Description: Lemma 2 for wwlkextbij 25473. (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 25469 . 2  |-  ( N  e.  NN0  ->  F : D
--> R )
5 fveq2 5870 . . . . . . 7  |-  ( t  =  d  ->  ( lastS  `  t )  =  ( lastS  `  d ) )
6 fvex 5880 . . . . . . 7  |-  ( lastS  `  d
)  e.  _V
75, 3, 6fvmpt 5953 . . . . . 6  |-  ( d  e.  D  ->  ( F `  d )  =  ( lastS  `  d ) )
8 fveq2 5870 . . . . . . 7  |-  ( t  =  x  ->  ( lastS  `  t )  =  ( lastS  `  x ) )
9 fvex 5880 . . . . . . 7  |-  ( lastS  `  x
)  e.  _V
108, 3, 9fvmpt 5953 . . . . . 6  |-  ( x  e.  D  ->  ( F `  x )  =  ( lastS  `  x ) )
117, 10eqeqan12d 2469 . . . . 5  |-  ( ( d  e.  D  /\  x  e.  D )  ->  ( ( F `  d )  =  ( F `  x )  <-> 
( lastS  `  d )  =  ( lastS  `  x )
) )
1211adantl 468 . . . 4  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( F `  d
)  =  ( F `
 x )  <->  ( lastS  `  d
)  =  ( lastS  `  x
) ) )
13 fveq2 5870 . . . . . . . . 9  |-  ( w  =  d  ->  ( # `
 w )  =  ( # `  d
) )
1413eqeq1d 2455 . . . . . . . 8  |-  ( w  =  d  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  d
)  =  ( N  +  2 ) ) )
15 oveq1 6302 . . . . . . . . 9  |-  ( w  =  d  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
d substr  <. 0 ,  ( N  +  1 )
>. ) )
1615eqeq1d 2455 . . . . . . . 8  |-  ( w  =  d  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
17 fveq2 5870 . . . . . . . . . 10  |-  ( w  =  d  ->  ( lastS  `  w )  =  ( lastS  `  d ) )
1817preq2d 4061 . . . . . . . . 9  |-  ( w  =  d  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  d
) } )
1918eleq1d 2515 . . . . . . . 8  |-  ( w  =  d  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  d
) }  e.  ran  E ) )
2014, 16, 193anbi123d 1341 . . . . . . 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 3200 . . . . . 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 5870 . . . . . . . . 9  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
2322eqeq1d 2455 . . . . . . . 8  |-  ( w  =  x  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  x
)  =  ( N  +  2 ) ) )
24 oveq1 6302 . . . . . . . . 9  |-  ( w  =  x  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. ) )
2524eqeq1d 2455 . . . . . . . 8  |-  ( w  =  x  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
26 fveq2 5870 . . . . . . . . . 10  |-  ( w  =  x  ->  ( lastS  `  w )  =  ( lastS  `  x ) )
2726preq2d 4061 . . . . . . . . 9  |-  ( w  =  x  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  x
) } )
2827eleq1d 2515 . . . . . . . 8  |-  ( w  =  x  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  x
) }  e.  ran  E ) )
2923, 25, 283anbi123d 1341 . . . . . . 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 3200 . . . . . 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 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( # `  x )  =  ( N  + 
2 ) )  -> 
( # `  d )  =  ( # `  x
) )
3231expcom 437 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( # `
 d )  =  ( # `  x
) ) )
33323ad2ant1 1030 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( ( # `
 d )  =  ( N  +  2 )  ->  ( # `  d
)  =  ( # `  x ) ) )
3433adantl 468 . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . 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 1030 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 431 . . . . . . . . . 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 467 . . . . . . . . 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 467 . . . . . . . 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 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 )
)  ->  ( lastS  `  d
)  =  ( lastS  `  x
) )
42 eqtr3 2474 . . . . . . . . . . . . . . . . . . . 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 10732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  1  =  ( 2  -  1 )
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  1  =  ( 2  -  1 ) )
4544oveq2d 6311 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( N  +  ( 2  -  1 ) ) )
46 nn0cn 10886 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  N  e.  CC )
47 2cnd 10689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  2  e.  CC )
48 1cnd 9664 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  1  e.  CC )
4946, 47, 48addsubassd 10011 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
5045, 49eqtr4d 2490 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( ( N  + 
2 )  -  1 ) )
5150adantr 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( N  +  1 )  =  ( ( N  +  2 )  -  1 ) )
52 oveq1 6302 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( # `  d )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
5352eqeq2d 2463 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( N  +  1 )  =  ( (
# `  d )  -  1 )  <->  ( N  +  1 )  =  ( ( N  + 
2 )  -  1 ) ) )
5453adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( ( N  + 
1 )  =  ( ( # `  d
)  -  1 )  <-> 
( N  +  1 )  =  ( ( N  +  2 )  -  1 ) ) )
5551, 54mpbird 236 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( N  +  1 )  =  ( (
# `  d )  -  1 ) )
56 opeq2 4170 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )
5756oveq2d 6311 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( d substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( d substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
5856oveq2d 6311 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
5957, 58eqeq12d 2468 . . . . . . . . . . . . . . . . . . . . . . . 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 211 . . . . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . . . . . 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 83 . . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . . 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 1031 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 32 . . . . . . . . . . . 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 1029 . . . . . . . . . . 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 468 . . . . . . . . . 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 434 . . . . . . . . 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 467 . . . . . . . 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 459 . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 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 570 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 10885 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  N  e.  RR )
81 2re 10686 . . . . . . . . . . . . . . . . . . . . . 22  |-  2  e.  RR
8281a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  2  e.  RR )
83 nn0ge0 10902 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <_  N )
84 2pos 10708 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  <  2
8584a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  <  2 )
8680, 82, 83, 85addgegt0d 10194 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
8786adantl 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
88 breq2 4409 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
0  <  ( # `  d
)  <->  0  <  ( N  +  2 ) ) )
8988adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
( 0  <  ( # `
 d )  <->  0  <  ( N  +  2 ) ) )
9087, 89mpbird 236 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( # `  d
) )
91 hashgt0n0 12553 . . . . . . . . . . . . . . . . . 18  |-  ( ( d  e. Word  V  /\  0  <  ( # `  d
) )  ->  d  =/=  (/) )
9290, 91sylan2 477 . . . . . . . . . . . . . . . . 17  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 ) )  ->  d  =/=  (/) )
9392exp32 610 . . . . . . . . . . . . . . . 16  |-  ( d  e. Word  V  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
9493com12 32 . . . . . . . . . . . . . . 15  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
d  e. Word  V  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
95943ad2ant1 1030 . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 431 . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
100 breq2 4409 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
0  <  ( # `  x
)  <->  0  <  ( N  +  2 ) ) )
101100adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
( 0  <  ( # `
 x )  <->  0  <  ( N  +  2 ) ) )
10299, 101mpbird 236 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( # `  x
) )
103 hashgt0n0 12553 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e. Word  V  /\  0  <  ( # `  x
) )  ->  x  =/=  (/) )
104102, 103sylan2 477 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 ) )  ->  x  =/=  (/) )
105104exp32 610 . . . . . . . . . . . . . . . 16  |-  ( x  e. Word  V  ->  (
( # `  x )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
106105com12 32 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
x  e. Word  V  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
1071063ad2ant1 1030 . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 431 . . . . . . . . . . 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 538 . . . . . . . . . 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 467 . . . . . . . . 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 459 . . . . . . . . . . . 12  |-  ( ( d  e. Word  V  /\  x  e. Word  V )  ->  d  e. Word  V )
114113adantr 467 . . . . . . . . . . 11  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  d  e. Word  V
)
115 simpr 463 . . . . . . . . . . . 12  |-  ( ( d  e. Word  V  /\  x  e. Word  V )  ->  x  e. Word  V )
116115adantr 467 . . . . . . . . . . 11  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  x  e. Word  V
)
117 hashneq0 12552 . . . . . . . . . . . . . . . 16  |-  ( d  e. Word  V  ->  (
0  <  ( # `  d
)  <->  d  =/=  (/) ) )
118117biimprd 227 . . . . . . . . . . . . . . 15  |-  ( d  e. Word  V  ->  (
d  =/=  (/)  ->  0  <  ( # `  d
) ) )
119118adantr 467 . . . . . . . . . . . . . 14  |-  ( ( d  e. Word  V  /\  x  e. Word  V )  ->  ( d  =/=  (/)  ->  0  <  ( # `  d
) ) )
120119com12 32 . . . . . . . . . . . . 13  |-  ( d  =/=  (/)  ->  ( (
d  e. Word  V  /\  x  e. Word  V )  ->  0  <  ( # `  d ) ) )
121120adantr 467 . . . . . . . . . . . 12  |-  ( ( d  =/=  (/)  /\  x  =/=  (/) )  ->  (
( d  e. Word  V  /\  x  e. Word  V )  ->  0  <  ( # `
 d ) ) )
122121impcom 432 . . . . . . . . . . 11  |-  ( ( ( d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) )  ->  0  <  ( # `
 d ) )
123 2swrd1eqwrdeq 12817 . . . . . . . . . . 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 1269 . . . . . . . . . 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 452 . . . . . . . . . . . 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 701 . . . . . . . . . . 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 990 . . . . . . . . . . 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 256 . . . . . . . . . 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 265 . . . . . . . . 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 1192 . . . . . . 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 609 . . . . . 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 482 . . . . 5  |-  ( ( d  e.  D  /\  x  e.  D )  ->  ( N  e.  NN0  ->  ( ( lastS  `  d
)  =  ( lastS  `  x
)  ->  d  =  x ) ) )
134133impcom 432 . . . 4  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( lastS  `  d )  =  ( lastS  `  x )  ->  d  =  x ) )
13512, 134sylbid 219 . . 3  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( F `  d
)  =  ( F `
 x )  -> 
d  =  x ) )
136135ralrimivva 2811 . 2  |-  ( N  e.  NN0  ->  A. d  e.  D  A. x  e.  D  ( ( F `  d )  =  ( F `  x )  ->  d  =  x ) )
137 dff13 6164 . 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 671 1  |-  ( N  e.  NN0  ->  F : D -1-1-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   {crab 2743   (/)c0 3733   {cpr 3972   <.cop 3976   class class class wbr 4405    |-> cmpt 4464   ran crn 4838   -->wf 5581   -1-1->wf1 5582   ` cfv 5585  (class class class)co 6295   RRcr 9543   0cc0 9544   1c1 9545    + caddc 9547    < clt 9680    - cmin 9865   2c2 10666   NN0cn0 10876   #chash 12522  Word cword 12663   lastS clsw 12664   substr csubstr 12667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-lsw 12672  df-s1 12674  df-substr 12675
This theorem is referenced by:  wwlkextbij0  25472
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