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Theorem wrdeqswrdlsw 12624
Description: Two words are equal iff they have the same length and the same prefix and the same last symbol. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Assertion
Ref Expression
wrdeqswrdlsw  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  (
# `  U )  /\  ( lastS  `  W )  =  ( lastS  `  U
)  /\  ( W substr  <.
0 ,  ( (
# `  W )  -  1 ) >.
)  =  ( U substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) ) ) )

Proof of Theorem wrdeqswrdlsw
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 eqwrd 12534 . . 3  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( W  =  U  <-> 
( ( # `  W
)  =  ( # `  U )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( U `  i
) ) ) )
21adantr 465 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  (
# `  U )  /\  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( U `  i
) ) ) )
3 0z 10864 . . . . . . 7  |-  0  e.  ZZ
4 lennncl 12516 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
5 elnnuz 11107 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN  <->  ( # `  W
)  e.  ( ZZ>= ` 
1 ) )
6 1e0p1 10993 . . . . . . . . . . . . 13  |-  1  =  ( 0  +  1 )
76fveq2i 5860 . . . . . . . . . . . 12  |-  ( ZZ>= ` 
1 )  =  (
ZZ>= `  ( 0  +  1 ) )
87eleq2i 2538 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  ( ZZ>= `  1 )  <->  (
# `  W )  e.  ( ZZ>= `  ( 0  +  1 ) ) )
95, 8sylbb 197 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( # `  W
)  e.  ( ZZ>= `  ( 0  +  1 ) ) )
104, 9syl 16 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ( ZZ>= `  ( 0  +  1 ) ) )
1110ad2ant2r 746 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( # `  W
)  e.  ( ZZ>= `  ( 0  +  1 ) ) )
1211adantr 465 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( # `
 W )  e.  ( ZZ>= `  ( 0  +  1 ) ) )
13 fzosplitsnm1 11847 . . . . . . 7  |-  ( ( 0  e.  ZZ  /\  ( # `  W )  e.  ( ZZ>= `  (
0  +  1 ) ) )  ->  (
0..^ ( # `  W
) )  =  ( ( 0..^ ( (
# `  W )  -  1 ) )  u.  { ( (
# `  W )  -  1 ) } ) )
143, 12, 13sylancr 663 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
0..^ ( # `  W
) )  =  ( ( 0..^ ( (
# `  W )  -  1 ) )  u.  { ( (
# `  W )  -  1 ) } ) )
1514raleqdv 3057 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( U `  i
)  <->  A. i  e.  ( ( 0..^ ( (
# `  W )  -  1 ) )  u.  { ( (
# `  W )  -  1 ) } ) ( W `  i )  =  ( U `  i ) ) )
16 ralunb 3678 . . . . 5  |-  ( A. i  e.  ( (
0..^ ( ( # `  W )  -  1 ) )  u.  {
( ( # `  W
)  -  1 ) } ) ( W `
 i )  =  ( U `  i
)  <->  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i )  /\  A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i ) ) )
1715, 16syl6bb 261 . . . 4  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( U `  i
)  <->  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i )  /\  A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i ) ) ) )
18 ovex 6300 . . . . . . . 8  |-  ( (
# `  W )  -  1 )  e. 
_V
19 fveq2 5857 . . . . . . . . . 10  |-  ( i  =  ( ( # `  W )  -  1 )  ->  ( W `  i )  =  ( W `  ( (
# `  W )  -  1 ) ) )
20 fveq2 5857 . . . . . . . . . 10  |-  ( i  =  ( ( # `  W )  -  1 )  ->  ( U `  i )  =  ( U `  ( (
# `  W )  -  1 ) ) )
2119, 20eqeq12d 2482 . . . . . . . . 9  |-  ( i  =  ( ( # `  W )  -  1 )  ->  ( ( W `  i )  =  ( U `  i )  <->  ( W `  ( ( # `  W
)  -  1 ) )  =  ( U `
 ( ( # `  W )  -  1 ) ) ) )
2221ralsng 4055 . . . . . . . 8  |-  ( ( ( # `  W
)  -  1 )  e.  _V  ->  ( A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i )  <-> 
( W `  (
( # `  W )  -  1 ) )  =  ( U `  ( ( # `  W
)  -  1 ) ) ) )
2318, 22mp1i 12 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i )  <-> 
( W `  (
( # `  W )  -  1 ) )  =  ( U `  ( ( # `  W
)  -  1 ) ) ) )
24 oveq1 6282 . . . . . . . . . 10  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( # `
 W )  - 
1 )  =  ( ( # `  U
)  -  1 ) )
2524fveq2d 5861 . . . . . . . . 9  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( U `  ( ( # `  W
)  -  1 ) )  =  ( U `
 ( ( # `  U )  -  1 ) ) )
2625eqeq2d 2474 . . . . . . . 8  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( W `  ( ( # `
 W )  - 
1 ) )  =  ( U `  (
( # `  W )  -  1 ) )  <-> 
( W `  (
( # `  W )  -  1 ) )  =  ( U `  ( ( # `  U
)  -  1 ) ) ) )
2726adantl 466 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( W `  (
( # `  W )  -  1 ) )  =  ( U `  ( ( # `  W
)  -  1 ) )  <->  ( W `  ( ( # `  W
)  -  1 ) )  =  ( U `
 ( ( # `  U )  -  1 ) ) ) )
28 lsw 12537 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
2928adantr 465 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( lastS  `  W )  =  ( W `  ( ( # `  W
)  -  1 ) ) )
3029eqcomd 2468 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( W `  (
( # `  W )  -  1 ) )  =  ( lastS  `  W
) )
31 lsw 12537 . . . . . . . . . . 11  |-  ( U  e. Word  V  ->  ( lastS  `  U )  =  ( U `  ( (
# `  U )  -  1 ) ) )
3231adantl 466 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( lastS  `  U )  =  ( U `  ( ( # `  U
)  -  1 ) ) )
3332eqcomd 2468 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( U `  (
( # `  U )  -  1 ) )  =  ( lastS  `  U
) )
3430, 33eqeq12d 2482 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( ( W `  ( ( # `  W
)  -  1 ) )  =  ( U `
 ( ( # `  U )  -  1 ) )  <->  ( lastS  `  W
)  =  ( lastS  `  U
) ) )
3534ad2antrr 725 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( W `  (
( # `  W )  -  1 ) )  =  ( U `  ( ( # `  U
)  -  1 ) )  <->  ( lastS  `  W )  =  ( lastS  `  U
) ) )
3623, 27, 353bitrd 279 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i )  <-> 
( lastS  `  W )  =  ( lastS  `  U )
) )
3736anbi2d 703 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i )  /\  A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i ) )  <->  ( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i )  /\  ( lastS  `  W )  =  ( lastS  `  U )
) ) )
38 ancom 450 . . . . . 6  |-  ( ( A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) ( W `
 i )  =  ( U `  i
)  /\  ( lastS  `  W
)  =  ( lastS  `  U
) )  <->  ( ( lastS  `  W )  =  ( lastS  `  U )  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) ( W `  i )  =  ( U `  i ) ) )
3938a1i 11 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i )  /\  ( lastS  `  W )  =  ( lastS  `  U )
)  <->  ( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) )
40 eqid 2460 . . . . . . . 8  |-  ( (
# `  W )  -  1 )  =  ( ( # `  W
)  -  1 )
4140biantrur 506 . . . . . . 7  |-  ( ( ( lastS  `  W )  =  ( lastS  `  U )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) )  <-> 
( ( ( # `  W )  -  1 )  =  ( (
# `  W )  -  1 )  /\  ( ( lastS  `  W )  =  ( lastS  `  U
)  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) )
4241a1i 11 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) )  <-> 
( ( ( # `  W )  -  1 )  =  ( (
# `  W )  -  1 )  /\  ( ( lastS  `  W )  =  ( lastS  `  U
)  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) ) )
43 an12 795 . . . . . 6  |-  ( ( ( ( # `  W
)  -  1 )  =  ( ( # `  W )  -  1 )  /\  ( ( lastS  `  W )  =  ( lastS  `  U )  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) ( W `  i )  =  ( U `  i ) ) )  <->  ( ( lastS  `  W )  =  ( lastS  `  U )  /\  (
( ( # `  W
)  -  1 )  =  ( ( # `  W )  -  1 )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) )
4442, 43syl6bb 261 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) )  <-> 
( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( (
( # `  W )  -  1 )  =  ( ( # `  W
)  -  1 )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) ) )
4537, 39, 443bitrd 279 . . . 4  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i )  /\  A. i  e.  { ( ( # `  W
)  -  1 ) }  ( W `  i )  =  ( U `  i ) )  <->  ( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( (
( # `  W )  -  1 )  =  ( ( # `  W
)  -  1 )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) ) )
46 simpll 753 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( W  e. Word  V  /\  U  e. Word  V ) )
47 nnm1nn0 10826 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
484, 47syl 16 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  (
( # `  W )  -  1 )  e. 
NN0 )
4948ad2ant2r 746 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( ( # `  W )  -  1 )  e.  NN0 )
5049adantr 465 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( # `  W )  -  1 )  e. 
NN0 )
51 lencl 12515 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
52 nn0re 10793 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
5352lem1d 10468 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  - 
1 )  <_  ( # `
 W ) )
5451, 53syl 16 . . . . . . . . 9  |-  ( W  e. Word  V  ->  (
( # `  W )  -  1 )  <_ 
( # `  W ) )
5554adantr 465 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( ( # `  W
)  -  1 )  <_  ( # `  W
) )
5655ad2antrr 725 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( # `  W )  -  1 )  <_ 
( # `  W ) )
57 breq2 4444 . . . . . . . . . 10  |-  ( (
# `  U )  =  ( # `  W
)  ->  ( (
( # `  W )  -  1 )  <_ 
( # `  U )  <-> 
( ( # `  W
)  -  1 )  <_  ( # `  W
) ) )
5857eqcoms 2472 . . . . . . . . 9  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( (
( # `  W )  -  1 )  <_ 
( # `  U )  <-> 
( ( # `  W
)  -  1 )  <_  ( # `  W
) ) )
5958adantl 466 . . . . . . . 8  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( ( # `  W
)  -  1 )  <_  ( # `  U
)  <->  ( ( # `  W )  -  1 )  <_  ( # `  W
) ) )
6056, 59mpbird 232 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( # `  W )  -  1 )  <_ 
( # `  U ) )
61 swrdeq 12621 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( ( (
# `  W )  -  1 )  e. 
NN0  /\  ( ( # `
 W )  - 
1 )  e.  NN0 )  /\  ( ( (
# `  W )  -  1 )  <_ 
( # `  W )  /\  ( ( # `  W )  -  1 )  <_  ( # `  U
) ) )  -> 
( ( W substr  <. 0 ,  ( ( # `  W )  -  1 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  <->  ( ( (
# `  W )  -  1 )  =  ( ( # `  W
)  -  1 )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) )
6246, 50, 50, 56, 60, 61syl122anc 1232 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  <->  ( ( (
# `  W )  -  1 )  =  ( ( # `  W
)  -  1 )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) ) )
6362bicomd 201 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( ( ( # `  W )  -  1 )  =  ( (
# `  W )  -  1 )  /\  A. i  e.  ( 0..^ ( ( # `  W
)  -  1 ) ) ( W `  i )  =  ( U `  i ) )  <->  ( W substr  <. 0 ,  ( ( # `  W )  -  1 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) ) )
6463anbi2d 703 . . . 4  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( (
( # `  W )  -  1 )  =  ( ( # `  W
)  -  1 )  /\  A. i  e.  ( 0..^ ( (
# `  W )  -  1 ) ) ( W `  i
)  =  ( U `
 i ) ) )  <->  ( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( W substr  <.
0 ,  ( (
# `  W )  -  1 ) >.
)  =  ( U substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) ) ) )
6517, 45, 643bitrd 279 . . 3  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( U `  i
)  <->  ( ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( W substr  <.
0 ,  ( (
# `  W )  -  1 ) >.
)  =  ( U substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) ) ) )
6665pm5.32da 641 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( ( (
# `  W )  =  ( # `  U
)  /\  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( U `  i ) )  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( ( lastS  `  W )  =  ( lastS  `  U )  /\  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) ) ) ) )
67 3anass 972 . . . 4  |-  ( ( ( # `  W
)  =  ( # `  U )  /\  ( lastS  `  W )  =  ( lastS  `  U )  /\  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) )  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( ( lastS  `  W )  =  ( lastS  `  U )  /\  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) ) ) )
6867bicomi 202 . . 3  |-  ( ( ( # `  W
)  =  ( # `  U )  /\  (
( lastS  `  W )  =  ( lastS  `  U )  /\  ( W substr  <. 0 ,  ( ( # `  W )  -  1 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) ) )  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( W substr  <.
0 ,  ( (
# `  W )  -  1 ) >.
)  =  ( U substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) ) )
6968a1i 11 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( ( (
# `  W )  =  ( # `  U
)  /\  ( ( lastS  `  W )  =  ( lastS  `  U )  /\  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) ) )  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( lastS  `  W
)  =  ( lastS  `  U
)  /\  ( W substr  <.
0 ,  ( (
# `  W )  -  1 ) >.
)  =  ( U substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) ) ) )
702, 66, 693bitrd 279 1  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( W  =/=  (/)  /\  U  =/=  (/) ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  (
# `  U )  /\  ( lastS  `  W )  =  ( lastS  `  U
)  /\  ( W substr  <.
0 ,  ( (
# `  W )  -  1 ) >.
)  =  ( U substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   _Vcvv 3106    u. cun 3467   (/)c0 3778   {csn 4020   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9794   NNcn 10525   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071  ..^cfzo 11781   #chash 12360  Word cword 12487   lastS clsw 12488   substr csubstr 12491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-lsw 12496  df-substr 12499
This theorem is referenced by:  wwlkextinj  24392
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