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Theorem wwlkextinj 24403
Description: Lemma 2 for wwlkextbij 24406. (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 24402 . 2  |-  ( N  e.  NN0  ->  F : D
--> R )
5 fvex 5874 . . . . . . 7  |-  ( lastS  `  d
)  e.  _V
6 fveq2 5864 . . . . . . . 8  |-  ( t  =  d  ->  ( lastS  `  t )  =  ( lastS  `  d ) )
76, 3fvmptg 5946 . . . . . . 7  |-  ( ( d  e.  D  /\  ( lastS  `  d )  e. 
_V )  ->  ( F `  d )  =  ( lastS  `  d ) )
85, 7mpan2 671 . . . . . 6  |-  ( d  e.  D  ->  ( F `  d )  =  ( lastS  `  d ) )
9 fvex 5874 . . . . . . 7  |-  ( lastS  `  x
)  e.  _V
10 fveq2 5864 . . . . . . . 8  |-  ( t  =  x  ->  ( lastS  `  t )  =  ( lastS  `  x ) )
1110, 3fvmptg 5946 . . . . . . 7  |-  ( ( x  e.  D  /\  ( lastS  `  x )  e. 
_V )  ->  ( F `  x )  =  ( lastS  `  x ) )
129, 11mpan2 671 . . . . . 6  |-  ( x  e.  D  ->  ( F `  x )  =  ( lastS  `  x ) )
138, 12eqeqan12d 2490 . . . . 5  |-  ( ( d  e.  D  /\  x  e.  D )  ->  ( ( F `  d )  =  ( F `  x )  <-> 
( lastS  `  d )  =  ( lastS  `  x )
) )
1413adantl 466 . . . 4  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( F `  d
)  =  ( F `
 x )  <->  ( lastS  `  d
)  =  ( lastS  `  x
) ) )
151eleq2i 2545 . . . . . . 7  |-  ( d  e.  D  <->  d  e.  { w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )
16 fveq2 5864 . . . . . . . . . 10  |-  ( w  =  d  ->  ( # `
 w )  =  ( # `  d
) )
1716eqeq1d 2469 . . . . . . . . 9  |-  ( w  =  d  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  d
)  =  ( N  +  2 ) ) )
18 oveq1 6289 . . . . . . . . . 10  |-  ( w  =  d  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
d substr  <. 0 ,  ( N  +  1 )
>. ) )
1918eqeq1d 2469 . . . . . . . . 9  |-  ( w  =  d  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
20 fveq2 5864 . . . . . . . . . . 11  |-  ( w  =  d  ->  ( lastS  `  w )  =  ( lastS  `  d ) )
2120preq2d 4113 . . . . . . . . . 10  |-  ( w  =  d  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  d
) } )
2221eleq1d 2536 . . . . . . . . 9  |-  ( w  =  d  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  d
) }  e.  ran  E ) )
2317, 19, 223anbi123d 1299 . . . . . . . 8  |-  ( 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
) ) )
2423elrab 3261 . . . . . . 7  |-  ( d  e.  { w  e. Word  V  |  ( ( # `
 w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  <->  ( d  e. Word  V  /\  ( (
# `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) ) )
2515, 24bitri 249 . . . . . 6  |-  ( d  e.  D  <->  ( d  e. Word  V  /\  ( (
# `  d )  =  ( N  + 
2 )  /\  (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) ) )
261eleq2i 2545 . . . . . . 7  |-  ( x  e.  D  <->  x  e.  { w  e. Word  V  | 
( ( # `  w
)  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  w
) }  e.  ran  E ) } )
27 fveq2 5864 . . . . . . . . . 10  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
2827eqeq1d 2469 . . . . . . . . 9  |-  ( w  =  x  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  x
)  =  ( N  +  2 ) ) )
29 oveq1 6289 . . . . . . . . . 10  |-  ( w  =  x  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. ) )
3029eqeq1d 2469 . . . . . . . . 9  |-  ( w  =  x  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
31 fveq2 5864 . . . . . . . . . . 11  |-  ( w  =  x  ->  ( lastS  `  w )  =  ( lastS  `  x ) )
3231preq2d 4113 . . . . . . . . . 10  |-  ( w  =  x  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  x
) } )
3332eleq1d 2536 . . . . . . . . 9  |-  ( w  =  x  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  x
) }  e.  ran  E ) )
3428, 30, 333anbi123d 1299 . . . . . . . 8  |-  ( 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 ) ) )
3534elrab 3261 . . . . . . 7  |-  ( x  e.  { w  e. Word  V  |  ( ( # `
 w )  =  ( N  +  2 )  /\  ( w substr  <. 0 ,  ( N  +  1 ) >.
)  =  W  /\  { ( lastS  `  W ) ,  ( lastS  `  w ) }  e.  ran  E
) }  <->  ( x  e. Word  V  /\  ( (
# `  x )  =  ( N  + 
2 )  /\  (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
3626, 35bitri 249 . . . . . 6  |-  ( x  e.  D  <->  ( x  e. Word  V  /\  ( (
# `  x )  =  ( N  + 
2 )  /\  (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) ) )
37 eqtr3 2495 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( # `  x )  =  ( N  + 
2 ) )  -> 
( # `  d )  =  ( # `  x
) )
3837expcom 435 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( # `
 d )  =  ( # `  x
) ) )
39383ad2ant1 1017 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( ( # `
 d )  =  ( N  +  2 )  ->  ( # `  d
)  =  ( # `  x ) ) )
4039adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( 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
) ) )
4140com12 31 . . . . . . . . . . . . . 14  |-  ( (
# `  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
) ) )
42413ad2ant1 1017 . . . . . . . . . . . . 13  |-  ( ( ( # `  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
) ) )
4342adantl 466 . . . . . . . . . . . 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 )  =  ( # `  x
) ) )
4443imp 429 . . . . . . . . . . 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
) )
4544adantr 465 . . . . . . . . . 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 )  =  ( # `  x
) )
4645adantr 465 . . . . . . . . 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
)  =  ( # `  x ) )
47 simpr 461 . . . . . . . . 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 )
)  ->  ( lastS  `  d
)  =  ( lastS  `  x
) )
48 eqtr3 2495 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( d substr  <. 0 ,  ( N  + 
1 ) >. )  =  W  /\  (
x substr  <. 0 ,  ( N  +  1 )
>. )  =  W
)  ->  ( d substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( N  +  1 ) >.
) )
49 1e2m1 10647 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  1  =  ( 2  -  1 )
5049a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  1  =  ( 2  -  1 ) )
5150oveq2d 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( N  +  ( 2  -  1 ) ) )
52 nn0cn 10801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  N  e.  CC )
53 2cnd 10604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  2  e.  CC )
54 ax-1cn 9546 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  1  e.  CC
5554a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  1  e.  CC )
5652, 53, 55addsubassd 9946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
5751, 56eqtr4d 2511 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( ( N  + 
2 )  -  1 ) )
5857adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( N  +  1 )  =  ( ( N  +  2 )  -  1 ) )
59 oveq1 6289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( # `  d )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
6059eqeq2d 2481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( N  +  1 )  =  ( (
# `  d )  -  1 )  <->  ( N  +  1 )  =  ( ( N  + 
2 )  -  1 ) ) )
6160adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( ( N  + 
1 )  =  ( ( # `  d
)  -  1 )  <-> 
( N  +  1 )  =  ( ( N  +  2 )  -  1 ) ) )
6258, 61mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  e.  NN0  /\  ( # `  d )  =  ( N  + 
2 ) )  -> 
( N  +  1 )  =  ( (
# `  d )  -  1 ) )
63 opeq2 4214 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )
6463oveq2d 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( d substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( d substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
6563oveq2d 6298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
6664, 65eqeq12d 2489 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( (
d substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. )  <->  ( d substr  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d )  -  1 ) >. ) ) )
6762, 66syl 16 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 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 ) >. ) ) )
6867biimpd 207 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 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 ) >.
) ) )
6968ex 434 . . . . . . . . . . . . . . . . . . . . . . 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 ) >.
) ) ) )
7069com13 80 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 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 )
>. ) ) ) )
7148, 70syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 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 ) >.
) ) ) )
7271ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 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 )
>. ) ) ) ) )
7372com23 78 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 ) >.
) ) ) ) )
7473impcom 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  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 )
>. ) ) ) )
7574com12 31 . . . . . . . . . . . . . . . . 17  |-  ( ( 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 ) >. ) ) ) )
76753ad2ant2 1018 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  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 ) >.
) ) ) )
7776adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( 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 ) >.
) ) ) )
7877com12 31 . . . . . . . . . . . . . 14  |-  ( ( ( # `  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 )
>. ) ) ) )
79783adant3 1016 . . . . . . . . . . . . 13  |-  ( ( ( # `  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 )
>. ) ) ) )
8079adantl 466 . . . . . . . . . . . 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 substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. )  =  (
x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) ) )
8180imp 429 . . . . . . . . . . 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 substr  <. 0 ,  ( ( # `  d )  -  1 ) >. )  =  ( x substr  <. 0 ,  ( ( # `  d
)  -  1 )
>. ) ) )
8281imp 429 . . . . . . . . . 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 )
>. ) )
8382adantr 465 . . . . . . . . 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 substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
84 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  ->  d  e. Word  V )
85 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  x  e. Word  V )
8684, 85anim12i 566 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) )
8786adantr 465 . . . . . . . . . . . 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  e. Word  V  /\  x  e. Word  V ) )
88 nn0re 10800 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
89 2re 10601 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  RR
9089a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  2  e.  RR )
91 nn0ge0 10817 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <_  N )
92 2pos 10623 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  <  2
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <  2 )
9488, 90, 91, 93addgegt0d 10122 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
9594adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
96 breq2 4451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
0  <  ( # `  d
)  <->  0  <  ( N  +  2 ) ) )
9796adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
( 0  <  ( # `
 d )  <->  0  <  ( N  +  2 ) ) )
9895, 97mpbird 232 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( # `  d
) )
99 hashgt0n0 12397 . . . . . . . . . . . . . . . . . . 19  |-  ( ( d  e. Word  V  /\  0  <  ( # `  d
) )  ->  d  =/=  (/) )
10098, 99sylan2 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 ) )  ->  d  =/=  (/) )
101100exp32 605 . . . . . . . . . . . . . . . . 17  |-  ( d  e. Word  V  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
102101com12 31 . . . . . . . . . . . . . . . 16  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
d  e. Word  V  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
1031023ad2ant1 1017 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E )  ->  ( d  e. Word  V  ->  ( N  e.  NN0  ->  d  =/=  (/) ) ) )
104103impcom 430 . . . . . . . . . . . . . 14  |-  ( ( d  e. Word  V  /\  ( ( # `  d
)  =  ( N  +  2 )  /\  ( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  d
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  d  =/=  (/) ) )
105104adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( 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  =/=  (/) ) )
106105imp 429 . . . . . . . . . . . 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  =/=  (/) )
10794adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
108 breq2 4451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
0  <  ( # `  x
)  <->  0  <  ( N  +  2 ) ) )
109108adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
( 0  <  ( # `
 x )  <->  0  <  ( N  +  2 ) ) )
110107, 109mpbird 232 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( # `  x
) )
111 hashgt0n0 12397 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e. Word  V  /\  0  <  ( # `  x
) )  ->  x  =/=  (/) )
112110, 111sylan2 474 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  N  e.  NN0 ) )  ->  x  =/=  (/) )
113112exp32 605 . . . . . . . . . . . . . . . . 17  |-  ( x  e. Word  V  ->  (
( # `  x )  =  ( N  + 
2 )  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
114113com12 31 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
x  e. Word  V  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
1151143ad2ant1 1017 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( x  e. Word  V  ->  ( N  e.  NN0  ->  x  =/=  (/) ) ) )
116115impcom 430 . . . . . . . . . . . . . 14  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  ( N  +  2 )  /\  ( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W  /\  { ( lastS  `  W
) ,  ( lastS  `  x
) }  e.  ran  E ) )  ->  ( N  e.  NN0  ->  x  =/=  (/) ) )
117116adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( 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  =/=  (/) ) )
118117imp 429 . . . . . . . . . . . 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  =/=  (/) )
11987, 106, 118jca32 535 . . . . . . . . . . 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
)  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) ) )
120119adantr 465 . . . . . . . . . 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 )  /\  ( lastS  `  d )  =  ( lastS  `  x )
)  ->  ( (
d  e. Word  V  /\  x  e. Word  V )  /\  ( d  =/=  (/)  /\  x  =/=  (/) ) ) )
121 wrdeqswrdlsw 12631 . . . . . . . . . 10  |-  ( ( ( 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 ) >.
) ) ) )
122120, 121syl 16 . . . . . . . . 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  =  x  <->  ( ( # `  d )  =  (
# `  x )  /\  ( lastS  `  d )  =  ( lastS  `  x
)  /\  ( d substr  <.
0 ,  ( (
# `  d )  -  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) ) ) )
12346, 47, 83, 122mpbir3and 1179 . . . . . . . 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 )
124123ex 434 . . . . . . 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 ) )
125124ex 434 . . . . . 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 ) ) )
12625, 36, 125syl2anb 479 . . . . 5  |-  ( ( d  e.  D  /\  x  e.  D )  ->  ( N  e.  NN0  ->  ( ( lastS  `  d
)  =  ( lastS  `  x
)  ->  d  =  x ) ) )
127126impcom 430 . . . 4  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( lastS  `  d )  =  ( lastS  `  x )  ->  d  =  x ) )
12814, 127sylbid 215 . . 3  |-  ( ( N  e.  NN0  /\  ( d  e.  D  /\  x  e.  D
) )  ->  (
( F `  d
)  =  ( F `
 x )  -> 
d  =  x ) )
129128ralrimivva 2885 . 2  |-  ( N  e.  NN0  ->  A. d  e.  D  A. x  e.  D  ( ( F `  d )  =  ( F `  x )  ->  d  =  x ) )
130 dff13 6152 . 2  |-  ( F : D -1-1-> R  <->  ( F : D --> R  /\  A. d  e.  D  A. x  e.  D  (
( F `  d
)  =  ( F `
 x )  -> 
d  =  x ) ) )
1314, 129, 130sylanbrc 664 1  |-  ( N  e.  NN0  ->  F : D -1-1-> R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113   (/)c0 3785   {cpr 4029   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   ran crn 5000   -->wf 5582   -1-1->wf1 5583   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    - cmin 9801   2c2 10581   NN0cn0 10791   #chash 12367  Word cword 12494   lastS clsw 12495   substr csubstr 12498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-lsw 12503  df-substr 12506
This theorem is referenced by:  wwlkextbij0  24405
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