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Theorem ccats1swrdeq 12362
Description: The last symbol of a word concatenated with the subword of the word having length less by 1 than the word results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.)
Assertion
Ref Expression
ccats1swrdeq  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  U  =  ( W concat  <" ( lastS  `  U ) "> ) ) )

Proof of Theorem ccats1swrdeq
StepHypRef Expression
1 oveq1 6097 . . . 4  |-  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  ( W concat  <" ( lastS  `  U
) "> )  =  ( ( U substr  <. 0 ,  ( # `  W ) >. ) concat  <" ( lastS  `  U ) "> ) )
21adantl 466 . . 3  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  -> 
( W concat  <" ( lastS  `  U ) "> )  =  ( ( U substr  <. 0 ,  (
# `  W ) >. ) concat  <" ( lastS  `  U
) "> )
)
3 oveq1 6097 . . . . . . . . . 10  |-  ( (
# `  U )  =  ( ( # `  W )  +  1 )  ->  ( ( # `
 U )  - 
1 )  =  ( ( ( # `  W
)  +  1 )  -  1 ) )
433ad2ant3 1011 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( ( # `
 U )  - 
1 )  =  ( ( ( # `  W
)  +  1 )  -  1 ) )
5 lencl 12248 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
6 nn0cn 10588 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  CC )
7 pncan1 9771 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  CC  ->  ( (
( # `  W )  +  1 )  - 
1 )  =  (
# `  W )
)
85, 6, 73syl 20 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  (
( ( # `  W
)  +  1 )  -  1 )  =  ( # `  W
) )
983ad2ant1 1009 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( (
( # `  W )  +  1 )  - 
1 )  =  (
# `  W )
)
104, 9eqtr2d 2475 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( # `  W
)  =  ( (
# `  U )  -  1 ) )
1110opeq2d 4065 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  <. 0 ,  ( # `  W
) >.  =  <. 0 ,  ( ( # `  U )  -  1 ) >. )
1211oveq2d 6106 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( U substr  <.
0 ,  ( # `  W ) >. )  =  ( U substr  <. 0 ,  ( ( # `  U )  -  1 ) >. ) )
1312oveq1d 6105 . . . . 5  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( ( U substr  <. 0 ,  (
# `  W ) >. ) concat  <" ( lastS  `  U
) "> )  =  ( ( U substr  <. 0 ,  ( (
# `  U )  -  1 ) >.
) concat  <" ( lastS  `  U
) "> )
)
14 simp2 989 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  U  e. Word  V )
15 nn0p1gt0 10608 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  0  <  ( ( # `  W
)  +  1 ) )
165, 15syl 16 . . . . . . . . 9  |-  ( W  e. Word  V  ->  0  <  ( ( # `  W
)  +  1 ) )
17163ad2ant1 1009 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  0  <  ( ( # `  W
)  +  1 ) )
18 breq2 4295 . . . . . . . . 9  |-  ( (
# `  U )  =  ( ( # `  W )  +  1 )  ->  ( 0  <  ( # `  U
)  <->  0  <  (
( # `  W )  +  1 ) ) )
19183ad2ant3 1011 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( 0  <  ( # `  U
)  <->  0  <  (
( # `  W )  +  1 ) ) )
2017, 19mpbird 232 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  0  <  (
# `  U )
)
21 hashneq0 12131 . . . . . . . 8  |-  ( U  e. Word  V  ->  (
0  <  ( # `  U
)  <->  U  =/=  (/) ) )
22213ad2ant2 1010 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( 0  <  ( # `  U
)  <->  U  =/=  (/) ) )
2320, 22mpbid 210 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  U  =/=  (/) )
24 swrdccatwrd 12361 . . . . . 6  |-  ( ( U  e. Word  V  /\  U  =/=  (/) )  ->  (
( U substr  <. 0 ,  ( ( # `  U
)  -  1 )
>. ) concat  <" ( lastS  `  U ) "> )  =  U )
2514, 23, 24syl2anc 661 . . . . 5  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( ( U substr  <. 0 ,  ( ( # `  U
)  -  1 )
>. ) concat  <" ( lastS  `  U ) "> )  =  U )
2613, 25eqtrd 2474 . . . 4  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( ( U substr  <. 0 ,  (
# `  W ) >. ) concat  <" ( lastS  `  U
) "> )  =  U )
2726adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  -> 
( ( U substr  <. 0 ,  ( # `  W
) >. ) concat  <" ( lastS  `  U ) "> )  =  U )
282, 27eqtr2d 2475 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  ->  U  =  ( W concat  <" ( lastS  `  U ) "> ) )
2928ex 434 1  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  U  =  ( W concat  <" ( lastS  `  U ) "> ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   (/)c0 3636   <.cop 3882   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   CCcc 9279   0cc0 9281   1c1 9282    + caddc 9284    < clt 9417    - cmin 9594   NN0cn0 10578   #chash 12102  Word cword 12220   lastS clsw 12221   concat cconcat 12222   <"cs1 12223   substr csubstr 12224
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-hash 12103  df-word 12228  df-lsw 12229  df-concat 12230  df-s1 12231  df-substr 12232
This theorem is referenced by:  ccats1swrdeqbi  12388  ccats1swrdeqrex  30257
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