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Theorem lswccat0lsw 12588
Description: The last symbol of a word concatenated with the empty word is the last symbol of the word. (Contributed by AV, 22-Oct-2018.)
Assertion
Ref Expression
lswccat0lsw  |-  ( W  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( lastS  `  W
) )

Proof of Theorem lswccat0lsw
StepHypRef Expression
1 wrd0 12546 . . . . . 6  |-  (/)  e. Word  V
2 ccatlen 12574 . . . . . 6  |-  ( ( W  e. Word  V  /\  (/) 
e. Word  V )  ->  ( # `
 ( W concat  (/) ) )  =  ( ( # `  W )  +  (
# `  (/) ) ) )
31, 2mpan2 671 . . . . 5  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  =  ( ( # `  W )  +  (
# `  (/) ) ) )
4 hash0 12417 . . . . . . 7  |-  ( # `  (/) )  =  0
54a1i 11 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 (/) )  =  0 )
65oveq2d 6311 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  +  ( # `  (/) ) )  =  ( ( # `  W )  +  0 ) )
7 lencl 12543 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
8 nn0cn 10817 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  CC )
97, 8syl 16 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
109addid1d 9791 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  +  0 )  =  ( # `  W
) )
113, 6, 103eqtrd 2512 . . . 4  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  =  ( # `  W
) )
1211oveq1d 6310 . . 3  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  =  ( (
# `  W )  -  1 ) )
1312fveq2d 5876 . 2  |-  ( W  e. Word  V  ->  ( W `  ( ( # `
 ( W concat  (/) ) )  -  1 ) )  =  ( W `  ( ( # `  W
)  -  1 ) ) )
14 ccatcl 12573 . . . . 5  |-  ( ( W  e. Word  V  /\  (/) 
e. Word  V )  ->  ( W concat 
(/) )  e. Word  V
)
151, 14mpan2 671 . . . 4  |-  ( W  e. Word  V  ->  ( W concat 
(/) )  e. Word  V
)
16 lsw 12565 . . . 4  |-  ( ( W concat  (/) )  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( ( W concat  (/) ) `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) ) )
1715, 16syl 16 . . 3  |-  ( W  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( ( W concat  (/) ) `  ( (
# `  ( W concat  (/) ) )  -  1 ) ) )
181a1i 11 . . . 4  |-  ( W  e. Word  V  ->  (/)  e. Word  V
)
19 nn0z 10899 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ZZ )
20 0z 10887 . . . . . . . . . 10  |-  0  e.  ZZ
214, 20eqeltri 2551 . . . . . . . . 9  |-  ( # `  (/) )  e.  ZZ
2221a1i 11 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  (/) )  e.  ZZ )
2319, 22zaddcld 10982 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  ( # `  (/) ) )  e.  ZZ )
247, 23syl 16 . . . . . 6  |-  ( W  e. Word  V  ->  (
( # `  W )  +  ( # `  (/) ) )  e.  ZZ )
253, 24eqeltrd 2555 . . . . 5  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  e.  ZZ )
26 peano2zm 10918 . . . . 5  |-  ( (
# `  ( W concat  (/) ) )  e.  ZZ  ->  ( ( # `  ( W concat 
(/) ) )  - 
1 )  e.  ZZ )
2725, 26syl 16 . . . 4  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  e.  ZZ )
28 ccatsymb 12580 . . . 4  |-  ( ( W  e. Word  V  /\  (/) 
e. Word  V  /\  (
( # `  ( W concat  (/) ) )  -  1 )  e.  ZZ )  ->  ( ( W concat  (/) ) `  ( (
# `  ( W concat  (/) ) )  -  1 ) )  =  if ( ( ( # `  ( W concat  (/) ) )  -  1 )  < 
( # `  W ) ,  ( W `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) ) ,  ( (/) `  ( ( ( # `  ( W concat 
(/) ) )  - 
1 )  -  ( # `
 W ) ) ) ) )
2918, 27, 28mpd3an23 1326 . . 3  |-  ( W  e. Word  V  ->  (
( W concat  (/) ) `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) )  =  if ( ( (
# `  ( W concat  (/) ) )  -  1 )  <  ( # `  W ) ,  ( W `  ( (
# `  ( W concat  (/) ) )  -  1 ) ) ,  (
(/) `  ( (
( # `  ( W concat  (/) ) )  -  1 )  -  ( # `  W ) ) ) ) )
303, 6eqtrd 2508 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  =  ( ( # `  W )  +  0 ) )
3130oveq1d 6310 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  =  ( ( ( # `  W
)  +  0 )  -  1 ) )
328addid1d 9791 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  0 )  =  (
# `  W )
)
3332oveq1d 6310 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  +  0 )  - 
1 )  =  ( ( # `  W
)  -  1 ) )
34 nn0re 10816 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
3534ltm1d 10490 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  - 
1 )  <  ( # `
 W ) )
3633, 35eqbrtrd 4473 . . . . . 6  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  +  0 )  - 
1 )  <  ( # `
 W ) )
377, 36syl 16 . . . . 5  |-  ( W  e. Word  V  ->  (
( ( # `  W
)  +  0 )  -  1 )  < 
( # `  W ) )
3831, 37eqbrtrd 4473 . . . 4  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  <  ( # `  W ) )
39 iftrue 3951 . . . 4  |-  ( ( ( # `  ( W concat 
(/) ) )  - 
1 )  <  ( # `
 W )  ->  if ( ( ( # `  ( W concat  (/) ) )  -  1 )  < 
( # `  W ) ,  ( W `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) ) ,  ( (/) `  ( ( ( # `  ( W concat 
(/) ) )  - 
1 )  -  ( # `
 W ) ) ) )  =  ( W `  ( (
# `  ( W concat  (/) ) )  -  1 ) ) )
4038, 39syl 16 . . 3  |-  ( W  e. Word  V  ->  if ( ( ( # `  ( W concat  (/) ) )  -  1 )  < 
( # `  W ) ,  ( W `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) ) ,  ( (/) `  ( ( ( # `  ( W concat 
(/) ) )  - 
1 )  -  ( # `
 W ) ) ) )  =  ( W `  ( (
# `  ( W concat  (/) ) )  -  1 ) ) )
4117, 29, 403eqtrd 2512 . 2  |-  ( W  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( W `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) ) )
42 lsw 12565 . 2  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
4313, 41, 423eqtr4d 2518 1  |-  ( W  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( lastS  `  W
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   (/)c0 3790   ifcif 3945   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    < clt 9640    - cmin 9817   NN0cn0 10807   ZZcz 10876   #chash 12385  Word cword 12515   lastS clsw 12516   concat cconcat 12517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-lsw 12524  df-concat 12525
This theorem is referenced by: (None)
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