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Theorem ccats1swrdeqrex 30259
Description: There exists a symbol such that its concatenation with the subword obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Proof shortened by AV, 24-Oct-2018.)
Assertion
Ref Expression
ccats1swrdeqrex  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  E. s  e.  V  U  =  ( W concat  <" s "> ) ) )
Distinct variable groups:    U, s    V, s    W, s

Proof of Theorem ccats1swrdeqrex
StepHypRef Expression
1 simp2 989 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  U  e. Word  V )
2 lencl 12249 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
3 nn0p1gt0 10609 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  0  <  ( ( # `  W
)  +  1 ) )
42, 3syl 16 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  0  <  ( ( # `  W
)  +  1 ) )
543ad2ant1 1009 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  0  <  ( ( # `  W
)  +  1 ) )
6 breq2 4296 . . . . . . . . . 10  |-  ( (
# `  U )  =  ( ( # `  W )  +  1 )  ->  ( 0  <  ( # `  U
)  <->  0  <  (
( # `  W )  +  1 ) ) )
763ad2ant3 1011 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( 0  <  ( # `  U
)  <->  0  <  (
( # `  W )  +  1 ) ) )
85, 7mpbird 232 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  0  <  (
# `  U )
)
9 hashneq0 12132 . . . . . . . . 9  |-  ( U  e. Word  V  ->  (
0  <  ( # `  U
)  <->  U  =/=  (/) ) )
1093ad2ant2 1010 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( 0  <  ( # `  U
)  <->  U  =/=  (/) ) )
118, 10mpbid 210 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  U  =/=  (/) )
121, 11jca 532 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( U  e. Word  V  /\  U  =/=  (/) ) )
1312adantr 465 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  -> 
( U  e. Word  V  /\  U  =/=  (/) ) )
14 lswcl 12270 . . . . 5  |-  ( ( U  e. Word  V  /\  U  =/=  (/) )  ->  ( lastS  `  U )  e.  V
)
1513, 14syl 16 . . . 4  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  -> 
( lastS  `  U )  e.  V )
16 ccats1swrdeq 12363 . . . . 5  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  U  =  ( W concat  <" ( lastS  `  U ) "> ) ) )
1716imp 429 . . . 4  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  ->  U  =  ( W concat  <" ( lastS  `  U ) "> ) )
1815, 17jca 532 . . 3  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  -> 
( ( lastS  `  U
)  e.  V  /\  U  =  ( W concat  <" ( lastS  `  U ) "> ) ) )
19 s1eq 12291 . . . . . 6  |-  ( s  =  ( lastS  `  U
)  ->  <" s ">  =  <" ( lastS  `  U ) "> )
2019oveq2d 6107 . . . . 5  |-  ( s  =  ( lastS  `  U
)  ->  ( W concat  <" s "> )  =  ( W concat  <" ( lastS  `  U ) "> ) )
2120eqeq2d 2454 . . . 4  |-  ( s  =  ( lastS  `  U
)  ->  ( U  =  ( W concat  <" s "> )  <->  U  =  ( W concat  <" ( lastS  `  U ) "> ) ) )
2221rspcev 3073 . . 3  |-  ( ( ( lastS  `  U )  e.  V  /\  U  =  ( W concat  <" ( lastS  `  U ) "> ) )  ->  E. s  e.  V  U  =  ( W concat  <" s "> ) )
2318, 22syl 16 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  ->  E. s  e.  V  U  =  ( W concat  <" s "> ) )
2423ex 434 1  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  E. s  e.  V  U  =  ( W concat  <" s "> ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   (/)c0 3637   <.cop 3883   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418   NN0cn0 10579   #chash 12103  Word cword 12221   lastS clsw 12222   concat cconcat 12223   <"cs1 12224   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-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-concat 12231  df-s1 12232  df-substr 12233
This theorem is referenced by:  ccats1rev  30260
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