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Theorem 2swrd1eqwrdeq 12670
Description: Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
Assertion
Ref Expression
2swrd1eqwrdeq  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( W  =  U  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )

Proof of Theorem 2swrd1eqwrdeq
StepHypRef Expression
1 lencl 12549 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
2 nn0z 10883 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ZZ )
3 elnnz 10870 . . . . . . . 8  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  ZZ  /\  0  <  ( # `  W ) ) )
43simplbi2 623 . . . . . . 7  |-  ( (
# `  W )  e.  ZZ  ->  ( 0  <  ( # `  W
)  ->  ( # `  W
)  e.  NN ) )
51, 2, 43syl 20 . . . . . 6  |-  ( W  e. Word  V  ->  (
0  <  ( # `  W
)  ->  ( # `  W
)  e.  NN ) )
65a1d 25 . . . . 5  |-  ( W  e. Word  V  ->  ( U  e. Word  V  ->  (
0  <  ( # `  W
)  ->  ( # `  W
)  e.  NN ) ) )
763imp 1188 . . . 4  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( # `
 W )  e.  NN )
8 fzo0end 11885 . . . 4  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  ( 0..^ ( # `  W
) ) )
97, 8syl 16 . . 3  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  (
( # `  W )  -  1 )  e.  ( 0..^ ( # `  W ) ) )
10 2swrdeqwrdeq 12669 . . 3  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  (
( # `  W )  -  1 )  e.  ( 0..^ ( # `  W ) ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  (
# `  U )  /\  ( ( W substr  <. 0 ,  ( ( # `  W )  -  1 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( W substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  ( U substr  <. (
( # `  W )  -  1 ) ,  ( # `  W
) >. ) ) ) ) )
119, 10syld3an3 1271 . 2  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( W  =  U  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( W substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  ( U substr  <. (
( # `  W )  -  1 ) ,  ( # `  W
) >. ) ) ) ) )
12 hashneq0 12417 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  (
0  <  ( # `  W
)  <->  W  =/=  (/) ) )
1312biimpd 207 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  (
0  <  ( # `  W
)  ->  W  =/=  (/) ) )
1413imdistani 688 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( W  e. Word  V  /\  W  =/=  (/) ) )
15143adant2 1013 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( W  e. Word  V  /\  W  =/=  (/) ) )
1615adantr 463 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( W  e. Word  V  /\  W  =/=  (/) ) )
17 swrdlsw 12668 . . . . . . 7  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( W substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  <" ( lastS  `  W
) "> )
1816, 17syl 16 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( W substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  <" ( lastS  `  W
) "> )
19 breq2 4443 . . . . . . . . . 10  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( 0  <  ( # `  W
)  <->  0  <  ( # `
 U ) ) )
20193anbi3d 1303 . . . . . . . . 9  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  < 
( # `  W ) )  <->  ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  U ) ) ) )
21 hashneq0 12417 . . . . . . . . . . . . 13  |-  ( U  e. Word  V  ->  (
0  <  ( # `  U
)  <->  U  =/=  (/) ) )
2221biimpd 207 . . . . . . . . . . . 12  |-  ( U  e. Word  V  ->  (
0  <  ( # `  U
)  ->  U  =/=  (/) ) )
2322imdistani 688 . . . . . . . . . . 11  |-  ( ( U  e. Word  V  /\  0  <  ( # `  U
) )  ->  ( U  e. Word  V  /\  U  =/=  (/) ) )
24233adant1 1012 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  U
) )  ->  ( U  e. Word  V  /\  U  =/=  (/) ) )
25 swrdlsw 12668 . . . . . . . . . 10  |-  ( ( U  e. Word  V  /\  U  =/=  (/) )  ->  ( U substr  <. ( ( # `  U )  -  1 ) ,  ( # `  U ) >. )  =  <" ( lastS  `  U
) "> )
2624, 25syl 16 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  U
) )  ->  ( U substr  <. ( ( # `  U )  -  1 ) ,  ( # `  U ) >. )  =  <" ( lastS  `  U
) "> )
2720, 26syl6bi 228 . . . . . . . 8  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  < 
( # `  W ) )  ->  ( U substr  <.
( ( # `  U
)  -  1 ) ,  ( # `  U
) >. )  =  <" ( lastS  `  U ) "> ) )
2827impcom 428 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( U substr  <. ( ( # `  U )  -  1 ) ,  ( # `  U ) >. )  =  <" ( lastS  `  U
) "> )
29 oveq1 6277 . . . . . . . . . . 11  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( # `
 W )  - 
1 )  =  ( ( # `  U
)  -  1 ) )
30 id 22 . . . . . . . . . . 11  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( # `  W
)  =  ( # `  U ) )
3129, 30opeq12d 4211 . . . . . . . . . 10  |-  ( (
# `  W )  =  ( # `  U
)  ->  <. ( (
# `  W )  -  1 ) ,  ( # `  W
) >.  =  <. (
( # `  U )  -  1 ) ,  ( # `  U
) >. )
3231oveq2d 6286 . . . . . . . . 9  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( U substr  <.
( ( # `  W
)  -  1 ) ,  ( # `  W
) >. )  =  ( U substr  <. ( ( # `  U )  -  1 ) ,  ( # `  U ) >. )
)
3332eqeq1d 2456 . . . . . . . 8  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( U substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  <" ( lastS  `  U
) ">  <->  ( U substr  <.
( ( # `  U
)  -  1 ) ,  ( # `  U
) >. )  =  <" ( lastS  `  U ) "> ) )
3433adantl 464 . . . . . . 7  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( U substr  <. ( (
# `  W )  -  1 ) ,  ( # `  W
) >. )  =  <" ( lastS  `  U ) ">  <->  ( U substr  <. (
( # `  U )  -  1 ) ,  ( # `  U
) >. )  =  <" ( lastS  `  U ) "> ) )
3528, 34mpbird 232 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( U substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  <" ( lastS  `  U
) "> )
3618, 35eqeq12d 2476 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( W substr  <. ( (
# `  W )  -  1 ) ,  ( # `  W
) >. )  =  ( U substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  <->  <" ( lastS  `  W ) ">  =  <" ( lastS  `  U ) "> ) )
37 hashgt0n0 12418 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  0  <  ( # `  W
) )  ->  W  =/=  (/) )
38 lswcl 12577 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  W  =/=  (/) )  ->  ( lastS  `  W )  e.  V
)
3937, 38syldan 468 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( lastS  `  W )  e.  V
)
40393adant2 1013 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( lastS  `  W )  e.  V
)
4140adantr 463 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( lastS  `  W )  e.  V
)
42 hashgt0n0 12418 . . . . . . . . . 10  |-  ( ( U  e. Word  V  /\  0  <  ( # `  U
) )  ->  U  =/=  (/) )
43 lswcl 12577 . . . . . . . . . 10  |-  ( ( U  e. Word  V  /\  U  =/=  (/) )  ->  ( lastS  `  U )  e.  V
)
4442, 43syldan 468 . . . . . . . . 9  |-  ( ( U  e. Word  V  /\  0  <  ( # `  U
) )  ->  ( lastS  `  U )  e.  V
)
45443adant1 1012 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  U
) )  ->  ( lastS  `  U )  e.  V
)
4620, 45syl6bi 228 . . . . . . 7  |-  ( (
# `  W )  =  ( # `  U
)  ->  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  < 
( # `  W ) )  ->  ( lastS  `  U
)  e.  V ) )
4746impcom 428 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( lastS  `  U )  e.  V
)
48 s111 12612 . . . . . 6  |-  ( ( ( lastS  `  W )  e.  V  /\  ( lastS  `  U )  e.  V
)  ->  ( <" ( lastS  `  W ) ">  =  <" ( lastS  `  U ) ">  <->  ( lastS  `  W )  =  ( lastS  `  U ) ) )
4941, 47, 48syl2anc 659 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  ( <" ( lastS  `  W
) ">  =  <" ( lastS  `  U
) ">  <->  ( lastS  `  W
)  =  ( lastS  `  U
) ) )
5036, 49bitrd 253 . . . 4  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( W substr  <. ( (
# `  W )  -  1 ) ,  ( # `  W
) >. )  =  ( U substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  <->  ( lastS  `  W )  =  ( lastS  `  U ) ) )
5150anbi2d 701 . . 3  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  /\  ( # `
 W )  =  ( # `  U
) )  ->  (
( ( W substr  <. 0 ,  ( ( # `  W )  -  1 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( W substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  ( U substr  <. (
( # `  W )  -  1 ) ,  ( # `  W
) >. ) )  <->  ( ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) )
5251pm5.32da 639 . 2  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  (
( ( # `  W
)  =  ( # `  U )  /\  (
( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( W substr  <. ( ( # `  W )  -  1 ) ,  ( # `  W ) >. )  =  ( U substr  <. (
( # `  W )  -  1 ) ,  ( # `  W
) >. ) ) )  <-> 
( ( # `  W
)  =  ( # `  U )  /\  (
( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
5311, 52bitrd 253 1  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  0  <  ( # `  W
) )  ->  ( W  =  U  <->  ( ( # `
 W )  =  ( # `  U
)  /\  ( ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( U substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   (/)c0 3783   <.cop 4022   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    < clt 9617    - cmin 9796   NNcn 10531   NN0cn0 10791   ZZcz 10860  ..^cfzo 11799   #chash 12387  Word cword 12518   lastS clsw 12519   <"cs1 12521   substr csubstr 12522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-lsw 12527  df-s1 12529  df-substr 12530
This theorem is referenced by:  wwlkextinj  24932  clwwlkf1  24998
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