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Theorem wwlkextinj 30362
Description: Lemma 2 for wwlkextbij 30365. (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 30361 . 2  |-  ( N  e.  NN0  ->  F : D
--> R )
5 fvex 5701 . . . . . . 7  |-  ( lastS  `  d
)  e.  _V
6 fveq2 5691 . . . . . . . 8  |-  ( t  =  d  ->  ( lastS  `  t )  =  ( lastS  `  d ) )
76, 3fvmptg 5772 . . . . . . 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 5701 . . . . . . 7  |-  ( lastS  `  x
)  e.  _V
10 fveq2 5691 . . . . . . . 8  |-  ( t  =  x  ->  ( lastS  `  t )  =  ( lastS  `  x ) )
1110, 3fvmptg 5772 . . . . . . 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 2458 . . . . 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 2507 . . . . . . 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 5691 . . . . . . . . . 10  |-  ( w  =  d  ->  ( # `
 w )  =  ( # `  d
) )
1716eqeq1d 2451 . . . . . . . . 9  |-  ( w  =  d  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  d
)  =  ( N  +  2 ) ) )
18 oveq1 6098 . . . . . . . . . 10  |-  ( w  =  d  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
d substr  <. 0 ,  ( N  +  1 )
>. ) )
1918eqeq1d 2451 . . . . . . . . 9  |-  ( w  =  d  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( d substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
20 fveq2 5691 . . . . . . . . . . 11  |-  ( w  =  d  ->  ( lastS  `  w )  =  ( lastS  `  d ) )
2120preq2d 3961 . . . . . . . . . 10  |-  ( w  =  d  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  d
) } )
2221eleq1d 2509 . . . . . . . . 9  |-  ( w  =  d  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  d
) }  e.  ran  E ) )
2317, 19, 223anbi123d 1289 . . . . . . . 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 3117 . . . . . . 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 2507 . . . . . . 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 5691 . . . . . . . . . 10  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
2827eqeq1d 2451 . . . . . . . . 9  |-  ( w  =  x  ->  (
( # `  w )  =  ( N  + 
2 )  <->  ( # `  x
)  =  ( N  +  2 ) ) )
29 oveq1 6098 . . . . . . . . . 10  |-  ( w  =  x  ->  (
w substr  <. 0 ,  ( N  +  1 )
>. )  =  (
x substr  <. 0 ,  ( N  +  1 )
>. ) )
3029eqeq1d 2451 . . . . . . . . 9  |-  ( w  =  x  ->  (
( w substr  <. 0 ,  ( N  +  1 ) >. )  =  W  <-> 
( x substr  <. 0 ,  ( N  +  1 ) >. )  =  W ) )
31 fveq2 5691 . . . . . . . . . . 11  |-  ( w  =  x  ->  ( lastS  `  w )  =  ( lastS  `  x ) )
3231preq2d 3961 . . . . . . . . . 10  |-  ( w  =  x  ->  { ( lastS  `  W ) ,  ( lastS  `  w ) }  =  { ( lastS  `  W ) ,  ( lastS  `  x
) } )
3332eleq1d 2509 . . . . . . . . 9  |-  ( w  =  x  ->  ( { ( lastS  `  W ) ,  ( lastS  `  w
) }  e.  ran  E  <->  { ( lastS  `  W ) ,  ( lastS  `  x
) }  e.  ran  E ) )
3428, 30, 333anbi123d 1289 . . . . . . . 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 3117 . . . . . . 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 2462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  ( # `  x )  =  ( N  + 
2 ) )  -> 
( # `  d )  =  ( # `  x
) )
3837expcom 435 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  x )  =  ( N  + 
2 )  ->  (
( # `  d )  =  ( N  + 
2 )  ->  ( # `
 d )  =  ( # `  x
) ) )
39383ad2ant1 1009 . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 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 2462 . . . . . . . . . . . . . . . . . . . . . 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 10437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  1  =  ( 2  -  1 )
5049a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  1  =  ( 2  -  1 ) )
5150oveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( N  +  ( 2  -  1 ) ) )
52 nn0cn 10589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  N  e.  CC )
53 2cnd 10394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  2  e.  CC )
54 ax-1cn 9340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  1  e.  CC
5554a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( N  e.  NN0  ->  1  e.  CC )
5652, 53, 55addsubassd 9739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  NN0  ->  ( ( N  +  2 )  -  1 )  =  ( N  +  ( 2  -  1 ) ) )
5751, 56eqtr4d 2478 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
# `  d )  =  ( N  + 
2 )  ->  (
( # `  d )  -  1 )  =  ( ( N  + 
2 )  -  1 ) )
6059eqeq2d 2454 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4060 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  <. 0 ,  ( N  +  1 ) >.  =  <. 0 ,  ( ( # `
 d )  - 
1 ) >. )
6463oveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( d substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( d substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
6563oveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( N  +  1 )  =  ( ( # `  d )  -  1 )  ->  ( x substr  <.
0 ,  ( N  +  1 ) >.
)  =  ( x substr  <. 0 ,  ( (
# `  d )  -  1 ) >.
) )
6664, 65eqeq12d 2457 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . 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 1008 . . . . . . . . . . . . 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 10588 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  RR )
89 2re 10391 . . . . . . . . . . . . . . . . . . . . . . 23  |-  2  e.  RR
9089a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  2  e.  RR )
91 nn0ge0 10605 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <_  N )
92 2pos 10413 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  <  2
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  0  <  2 )
9488, 90, 91, 93addgegt0d 9913 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  0  < 
( N  +  2 ) )
9594adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( # `  d
)  =  ( N  +  2 )  /\  N  e.  NN0 )  -> 
0  <  ( N  +  2 ) )
96 breq2 4296 . . . . . . . . . . . . . . . . . . . . 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 12133 . . . . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . 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 4296 . . . . . . . . . . . . . . . . . . . . 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 12133 . . . . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . 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 12343 . . . . . . . . . 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 1171 . . . . . . . 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 2808 . 2  |-  ( N  e.  NN0  ->  A. d  e.  D  A. x  e.  D  ( ( F `  d )  =  ( F `  x )  ->  d  =  x ) )
130 dff13 5971 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719   _Vcvv 2972   (/)c0 3637   {cpr 3879   <.cop 3883   class class class wbr 4292    e. cmpt 4350   ran crn 4841   -->wf 5414   -1-1->wf1 5415   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    - cmin 9595   2c2 10371   NN0cn0 10579   #chash 12103  Word cword 12221   lastS clsw 12222   substr csubstr 12225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-substr 12233
This theorem is referenced by:  wwlkextbij0  30364
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