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Theorem ccatval1lsw 12382
Description: The last symbol of the left (nonempty) half of a concatenated word. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
Assertion
Ref Expression
ccatval1lsw  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  A  =/=  (/) )  ->  (
( A concat  B ) `  ( ( # `  A
)  -  1 ) )  =  ( lastS  `  A
) )

Proof of Theorem ccatval1lsw
StepHypRef Expression
1 hashneq0 12230 . . . . . . 7  |-  ( A  e. Word  V  ->  (
0  <  ( # `  A
)  <->  A  =/=  (/) ) )
2 lencl 12348 . . . . . . . 8  |-  ( A  e. Word  V  ->  ( # `
 A )  e. 
NN0 )
3 nn0z 10767 . . . . . . . 8  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  ZZ )
4 elnnz 10754 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN  <->  ( ( # `  A )  e.  ZZ  /\  0  <  ( # `  A ) ) )
54simplbi2 625 . . . . . . . 8  |-  ( (
# `  A )  e.  ZZ  ->  ( 0  <  ( # `  A
)  ->  ( # `  A
)  e.  NN ) )
62, 3, 53syl 20 . . . . . . 7  |-  ( A  e. Word  V  ->  (
0  <  ( # `  A
)  ->  ( # `  A
)  e.  NN ) )
71, 6sylbird 235 . . . . . 6  |-  ( A  e. Word  V  ->  ( A  =/=  (/)  ->  ( # `  A
)  e.  NN ) )
87a1d 25 . . . . 5  |-  ( A  e. Word  V  ->  ( B  e. Word  V  ->  ( A  =/=  (/)  ->  ( # `  A
)  e.  NN ) ) )
983imp 1182 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  A  =/=  (/) )  ->  ( # `
 A )  e.  NN )
10 fzo0end 11717 . . . 4  |-  ( (
# `  A )  e.  NN  ->  ( ( # `
 A )  - 
1 )  e.  ( 0..^ ( # `  A
) ) )
119, 10syl 16 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  A  =/=  (/) )  ->  (
( # `  A )  -  1 )  e.  ( 0..^ ( # `  A ) ) )
12 ccatval1 12375 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  (
( # `  A )  -  1 )  e.  ( 0..^ ( # `  A ) ) )  ->  ( ( A concat  B ) `  (
( # `  A )  -  1 ) )  =  ( A `  ( ( # `  A
)  -  1 ) ) )
1311, 12syld3an3 1264 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  A  =/=  (/) )  ->  (
( A concat  B ) `  ( ( # `  A
)  -  1 ) )  =  ( A `
 ( ( # `  A )  -  1 ) ) )
14 lsw 12365 . . 3  |-  ( A  e. Word  V  ->  ( lastS  `  A )  =  ( A `  ( (
# `  A )  -  1 ) ) )
15143ad2ant1 1009 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  A  =/=  (/) )  ->  ( lastS  `  A )  =  ( A `  ( (
# `  A )  -  1 ) ) )
1613, 15eqtr4d 2494 1  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  A  =/=  (/) )  ->  (
( A concat  B ) `  ( ( # `  A
)  -  1 ) )  =  ( lastS  `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642   (/)c0 3732   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   0cc0 9380   1c1 9381    < clt 9516    - cmin 9693   NNcn 10420   NN0cn0 10677   ZZcz 10744  ..^cfzo 11646   #chash 12201  Word cword 12320   lastS clsw 12321   concat cconcat 12322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-card 8207  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-hash 12202  df-word 12328  df-lsw 12329  df-concat 12330
This theorem is referenced by:  clwwlkext2edg  30599  wwlkext2clwwlk  30600  clwwlkextfrlem1  30804
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