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Theorem ccats1swrdeqrex 12716
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 ++  <" s "> ) ) )
Distinct variable groups:    U, s    V, s    W, s

Proof of Theorem ccats1swrdeqrex
StepHypRef Expression
1 simp2 997 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  U  e. Word  V )
2 lencl 12569 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
3 nn0p1gt0 10846 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  0  <  ( ( # `  W
)  +  1 ) )
42, 3syl 16 . . . . . . . . . 10  |-  ( W  e. Word  V  ->  0  <  ( ( # `  W
)  +  1 ) )
543ad2ant1 1017 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  0  <  ( ( # `  W
)  +  1 ) )
6 breq2 4460 . . . . . . . . . 10  |-  ( (
# `  U )  =  ( ( # `  W )  +  1 )  ->  ( 0  <  ( # `  U
)  <->  0  <  (
( # `  W )  +  1 ) ) )
763ad2ant3 1019 . . . . . . . . 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 12437 . . . . . . . . 9  |-  ( U  e. Word  V  ->  (
0  <  ( # `  U
)  <->  U  =/=  (/) ) )
1093ad2ant2 1018 . . . . . . . 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 12597 . . . . 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 12706 . . . . 5  |-  ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `
 U )  =  ( ( # `  W
)  +  1 ) )  ->  ( W  =  ( U substr  <. 0 ,  ( # `  W
) >. )  ->  U  =  ( W ++  <" ( lastS  `  U ) "> ) ) )
1716imp 429 . . . 4  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V  /\  ( # `  U )  =  ( ( # `  W )  +  1 ) )  /\  W  =  ( U substr  <. 0 ,  ( # `  W
) >. ) )  ->  U  =  ( W ++  <" ( 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 ++  <" ( lastS  `  U ) "> ) ) )
19 s1eq 12621 . . . . . 6  |-  ( s  =  ( lastS  `  U
)  ->  <" s ">  =  <" ( lastS  `  U ) "> )
2019oveq2d 6312 . . . . 5  |-  ( s  =  ( lastS  `  U
)  ->  ( W ++  <" s "> )  =  ( W ++  <" ( lastS  `  U ) "> ) )
2120eqeq2d 2471 . . . 4  |-  ( s  =  ( lastS  `  U
)  ->  ( U  =  ( W ++  <" s "> )  <->  U  =  ( W ++  <" ( lastS  `  U ) "> ) ) )
2221rspcev 3210 . . 3  |-  ( ( ( lastS  `  U )  e.  V  /\  U  =  ( W ++  <" ( lastS  `  U ) "> ) )  ->  E. s  e.  V  U  =  ( W ++  <" 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 ++  <" 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 ++  <" s "> ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   (/)c0 3793   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    < clt 9645   NN0cn0 10816   #chash 12408  Word cword 12538   lastS clsw 12539   ++ cconcat 12540   <"cs1 12541   substr csubstr 12542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-lsw 12547  df-concat 12548  df-s1 12549  df-substr 12550
This theorem is referenced by:  reuccats1lem  12717
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