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Theorem lswccat0lsw 12392
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 12356 . . . . . 6  |-  (/)  e. Word  V
2 ccatlen 12379 . . . . . 6  |-  ( ( W  e. Word  V  /\  (/) 
e. Word  V )  ->  ( # `
 ( W concat  (/) ) )  =  ( ( # `  W )  +  (
# `  (/) ) ) )
31, 2mpan2 671 . . . . 5  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  =  ( ( # `  W )  +  (
# `  (/) ) ) )
4 hash0 12238 . . . . . . 7  |-  ( # `  (/) )  =  0
54a1i 11 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 (/) )  =  0 )
65oveq2d 6208 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  +  ( # `  (/) ) )  =  ( ( # `  W )  +  0 ) )
7 lencl 12353 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
8 nn0cn 10692 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  CC )
97, 8syl 16 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
109addid1d 9672 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  +  0 )  =  ( # `  W
) )
113, 6, 103eqtrd 2496 . . . 4  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  =  ( # `  W
) )
1211oveq1d 6207 . . 3  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  =  ( (
# `  W )  -  1 ) )
1312fveq2d 5795 . 2  |-  ( W  e. Word  V  ->  ( W `  ( ( # `
 ( W concat  (/) ) )  -  1 ) )  =  ( W `  ( ( # `  W
)  -  1 ) ) )
14 ccatcl 12378 . . . . 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 12370 . . . 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 10772 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ZZ )
20 0z 10760 . . . . . . . . . 10  |-  0  e.  ZZ
214, 20eqeltri 2535 . . . . . . . . 9  |-  ( # `  (/) )  e.  ZZ
2221a1i 11 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  (/) )  e.  ZZ )
2319, 22zaddcld 10854 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  ( # `  (/) ) )  e.  ZZ )
247, 23syl 16 . . . . . 6  |-  ( W  e. Word  V  ->  (
( # `  W )  +  ( # `  (/) ) )  e.  ZZ )
253, 24eqeltrd 2539 . . . . 5  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  e.  ZZ )
26 peano2zm 10791 . . . . 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 12385 . . . 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 1317 . . 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 2492 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 ( W concat  (/) ) )  =  ( ( # `  W )  +  0 ) )
3130oveq1d 6207 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  =  ( ( ( # `  W
)  +  0 )  -  1 ) )
328addid1d 9672 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  +  0 )  =  (
# `  W )
)
3332oveq1d 6207 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  +  0 )  - 
1 )  =  ( ( # `  W
)  -  1 ) )
34 nn0re 10691 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
3534ltm1d 10368 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  - 
1 )  <  ( # `
 W ) )
3633, 35eqbrtrd 4412 . . . . . 6  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  +  0 )  - 
1 )  <  ( # `
 W ) )
377, 36syl 16 . . . . 5  |-  ( W  e. Word  V  ->  (
( ( # `  W
)  +  0 )  -  1 )  < 
( # `  W ) )
3831, 37eqbrtrd 4412 . . . 4  |-  ( W  e. Word  V  ->  (
( # `  ( W concat  (/) ) )  -  1 )  <  ( # `  W ) )
39 iftrue 3897 . . . 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 2496 . 2  |-  ( W  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( W `  ( ( # `  ( W concat 
(/) ) )  - 
1 ) ) )
42 lsw 12370 . 2  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
4313, 41, 423eqtr4d 2502 1  |-  ( W  e. Word  V  ->  ( lastS  `  ( W concat  (/) ) )  =  ( lastS  `  W
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   (/)c0 3737   ifcif 3891   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   CCcc 9383   0cc0 9385   1c1 9386    + caddc 9388    < clt 9521    - cmin 9698   NN0cn0 10682   ZZcz 10749   #chash 12206  Word cword 12325   lastS clsw 12326   concat cconcat 12327
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
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 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-lsw 12334  df-concat 12335
This theorem is referenced by: (None)
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