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Theorem lswccatn0lsw 12409
Description: The last symbol of a word concatenated with a nonempty word is the last symbol of the nonempty word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 24-Nov-2018.)
Assertion
Ref Expression
lswccatn0lsw  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  ( lastS  `  ( A concat  B ) )  =  ( lastS  `  B
) )

Proof of Theorem lswccatn0lsw
StepHypRef Expression
1 ovex 6228 . . 3  |-  ( A concat  B )  e.  _V
2 lsw 12388 . . 3  |-  ( ( A concat  B )  e. 
_V  ->  ( lastS  `  ( A concat  B ) )  =  ( ( A concat  B
) `  ( ( # `
 ( A concat  B
) )  -  1 ) ) )
31, 2mp1i 12 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  ( lastS  `  ( A concat  B ) )  =  ( ( A concat  B ) `  ( ( # `  ( A concat  B ) )  - 
1 ) ) )
4 ccatcl 12396 . . . . . . 7  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( A concat  B )  e. Word  V )
5 lencl 12371 . . . . . . 7  |-  ( ( A concat  B )  e. Word  V  ->  ( # `  ( A concat  B ) )  e. 
NN0 )
64, 5syl 16 . . . . . 6  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A concat  B ) )  e. 
NN0 )
76nn0zd 10860 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A concat  B ) )  e.  ZZ )
8 peano2zm 10803 . . . . 5  |-  ( (
# `  ( A concat  B ) )  e.  ZZ  ->  ( ( # `  ( A concat  B ) )  - 
1 )  e.  ZZ )
97, 8syl 16 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( # `  ( A concat  B ) )  - 
1 )  e.  ZZ )
1093adant3 1008 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  (
( # `  ( A concat  B ) )  - 
1 )  e.  ZZ )
11 ccatsymb 12403 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  (
( # `  ( A concat  B ) )  - 
1 )  e.  ZZ )  ->  ( ( A concat  B ) `  (
( # `  ( A concat  B ) )  - 
1 ) )  =  if ( ( (
# `  ( A concat  B ) )  -  1 )  <  ( # `  A ) ,  ( A `  ( (
# `  ( A concat  B ) )  -  1 ) ) ,  ( B `  ( ( ( # `  ( A concat  B ) )  - 
1 )  -  ( # `
 A ) ) ) ) )
1210, 11syld3an3 1264 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  (
( A concat  B ) `  ( ( # `  ( A concat  B ) )  - 
1 ) )  =  if ( ( (
# `  ( A concat  B ) )  -  1 )  <  ( # `  A ) ,  ( A `  ( (
# `  ( A concat  B ) )  -  1 ) ) ,  ( B `  ( ( ( # `  ( A concat  B ) )  - 
1 )  -  ( # `
 A ) ) ) ) )
13 ccatlen 12397 . . . . . . . 8  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
1413oveq1d 6218 . . . . . . 7  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( # `  ( A concat  B ) )  - 
1 )  =  ( ( ( # `  A
)  +  ( # `  B ) )  - 
1 ) )
1514oveq1d 6218 . . . . . 6  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( ( # `  ( A concat  B ) )  -  1 )  -  ( # `  A
) )  =  ( ( ( ( # `  A )  +  (
# `  B )
)  -  1 )  -  ( # `  A
) ) )
16 lencl 12371 . . . . . . 7  |-  ( A  e. Word  V  ->  ( # `
 A )  e. 
NN0 )
17 lencl 12371 . . . . . . 7  |-  ( B  e. Word  V  ->  ( # `
 B )  e. 
NN0 )
18 nn0cn 10704 . . . . . . . 8  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  CC )
19 nn0cn 10704 . . . . . . . 8  |-  ( (
# `  B )  e.  NN0  ->  ( # `  B
)  e.  CC )
20 addcl 9479 . . . . . . . . . 10  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( # `  A
)  +  ( # `  B ) )  e.  CC )
21 ax-1cn 9455 . . . . . . . . . . 11  |-  1  e.  CC
2221a1i 11 . . . . . . . . . 10  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
1  e.  CC )
23 simpl 457 . . . . . . . . . 10  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( # `  A )  e.  CC )
2420, 22, 23sub32d 9866 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( ( (
# `  A )  +  ( # `  B
) )  -  1 )  -  ( # `  A ) )  =  ( ( ( (
# `  A )  +  ( # `  B
) )  -  ( # `
 A ) )  -  1 ) )
25 pncan2 9732 . . . . . . . . . 10  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( ( # `  A )  +  (
# `  B )
)  -  ( # `  A ) )  =  ( # `  B
) )
2625oveq1d 6218 . . . . . . . . 9  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( ( (
# `  A )  +  ( # `  B
) )  -  ( # `
 A ) )  -  1 )  =  ( ( # `  B
)  -  1 ) )
2724, 26eqtrd 2495 . . . . . . . 8  |-  ( ( ( # `  A
)  e.  CC  /\  ( # `  B )  e.  CC )  -> 
( ( ( (
# `  A )  +  ( # `  B
) )  -  1 )  -  ( # `  A ) )  =  ( ( # `  B
)  -  1 ) )
2818, 19, 27syl2an 477 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN0  /\  ( # `  B )  e.  NN0 )  -> 
( ( ( (
# `  A )  +  ( # `  B
) )  -  1 )  -  ( # `  A ) )  =  ( ( # `  B
)  -  1 ) )
2916, 17, 28syl2an 477 . . . . . 6  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( ( (
# `  A )  +  ( # `  B
) )  -  1 )  -  ( # `  A ) )  =  ( ( # `  B
)  -  1 ) )
3015, 29eqtrd 2495 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( ( # `  ( A concat  B ) )  -  1 )  -  ( # `  A
) )  =  ( ( # `  B
)  -  1 ) )
31303adant3 1008 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  (
( ( # `  ( A concat  B ) )  - 
1 )  -  ( # `
 A ) )  =  ( ( # `  B )  -  1 ) )
3231fveq2d 5806 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  ( B `  ( (
( # `  ( A concat  B ) )  - 
1 )  -  ( # `
 A ) ) )  =  ( B `
 ( ( # `  B )  -  1 ) ) )
33 lennncl 12372 . . . . . . . 8  |-  ( ( B  e. Word  V  /\  B  =/=  (/) )  ->  ( # `
 B )  e.  NN )
34 nnnlt1 10467 . . . . . . . . . . . 12  |-  ( (
# `  B )  e.  NN  ->  -.  ( # `
 B )  <  1 )
3534adantr 465 . . . . . . . . . . 11  |-  ( ( ( # `  B
)  e.  NN  /\  ( # `  A )  e.  NN0 )  ->  -.  ( # `  B
)  <  1 )
36 nn0re 10703 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  RR )
3736adantl 466 . . . . . . . . . . . 12  |-  ( ( ( # `  B
)  e.  NN  /\  ( # `  A )  e.  NN0 )  -> 
( # `  A )  e.  RR )
38 nnre 10444 . . . . . . . . . . . . 13  |-  ( (
# `  B )  e.  NN  ->  ( # `  B
)  e.  RR )
3938adantr 465 . . . . . . . . . . . 12  |-  ( ( ( # `  B
)  e.  NN  /\  ( # `  A )  e.  NN0 )  -> 
( # `  B )  e.  RR )
40 1red 9516 . . . . . . . . . . . 12  |-  ( ( ( # `  B
)  e.  NN  /\  ( # `  A )  e.  NN0 )  -> 
1  e.  RR )
41 ltaddsublt 10078 . . . . . . . . . . . 12  |-  ( ( ( # `  A
)  e.  RR  /\  ( # `  B )  e.  RR  /\  1  e.  RR )  ->  (
( # `  B )  <  1  <->  ( (
( # `  A )  +  ( # `  B
) )  -  1 )  <  ( # `  A ) ) )
4237, 39, 40, 41syl3anc 1219 . . . . . . . . . . 11  |-  ( ( ( # `  B
)  e.  NN  /\  ( # `  A )  e.  NN0 )  -> 
( ( # `  B
)  <  1  <->  ( (
( # `  A )  +  ( # `  B
) )  -  1 )  <  ( # `  A ) ) )
4335, 42mtbid 300 . . . . . . . . . 10  |-  ( ( ( # `  B
)  e.  NN  /\  ( # `  A )  e.  NN0 )  ->  -.  ( ( ( # `  A )  +  (
# `  B )
)  -  1 )  <  ( # `  A
) )
4416, 43sylan2 474 . . . . . . . . 9  |-  ( ( ( # `  B
)  e.  NN  /\  A  e. Word  V )  ->  -.  ( ( (
# `  A )  +  ( # `  B
) )  -  1 )  <  ( # `  A ) )
4544ex 434 . . . . . . . 8  |-  ( (
# `  B )  e.  NN  ->  ( A  e. Word  V  ->  -.  (
( ( # `  A
)  +  ( # `  B ) )  - 
1 )  <  ( # `
 A ) ) )
4633, 45syl 16 . . . . . . 7  |-  ( ( B  e. Word  V  /\  B  =/=  (/) )  ->  ( A  e. Word  V  ->  -.  ( ( ( # `  A )  +  (
# `  B )
)  -  1 )  <  ( # `  A
) ) )
4746com12 31 . . . . . 6  |-  ( A  e. Word  V  ->  (
( B  e. Word  V  /\  B  =/=  (/) )  ->  -.  ( ( ( # `  A )  +  (
# `  B )
)  -  1 )  <  ( # `  A
) ) )
48473impib 1186 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  -.  ( ( ( # `  A )  +  (
# `  B )
)  -  1 )  <  ( # `  A
) )
4914breq1d 4413 . . . . . . 7  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( ( # `  ( A concat  B ) )  -  1 )  <  ( # `  A
)  <->  ( ( (
# `  A )  +  ( # `  B
) )  -  1 )  <  ( # `  A ) ) )
5049notbid 294 . . . . . 6  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( -.  ( (
# `  ( A concat  B ) )  -  1 )  <  ( # `  A )  <->  -.  (
( ( # `  A
)  +  ( # `  B ) )  - 
1 )  <  ( # `
 A ) ) )
51503adant3 1008 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  ( -.  ( ( # `  ( A concat  B ) )  - 
1 )  <  ( # `
 A )  <->  -.  (
( ( # `  A
)  +  ( # `  B ) )  - 
1 )  <  ( # `
 A ) ) )
5248, 51mpbird 232 . . . 4  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  -.  ( ( # `  ( A concat  B ) )  - 
1 )  <  ( # `
 A ) )
53 iffalse 3910 . . . 4  |-  ( -.  ( ( # `  ( A concat  B ) )  - 
1 )  <  ( # `
 A )  ->  if ( ( ( # `  ( A concat  B ) )  -  1 )  <  ( # `  A
) ,  ( A `
 ( ( # `  ( A concat  B ) )  -  1 ) ) ,  ( B `
 ( ( (
# `  ( A concat  B ) )  -  1 )  -  ( # `  A ) ) ) )  =  ( B `
 ( ( (
# `  ( A concat  B ) )  -  1 )  -  ( # `  A ) ) ) )
5452, 53syl 16 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  if ( ( ( # `  ( A concat  B ) )  -  1 )  <  ( # `  A
) ,  ( A `
 ( ( # `  ( A concat  B ) )  -  1 ) ) ,  ( B `
 ( ( (
# `  ( A concat  B ) )  -  1 )  -  ( # `  A ) ) ) )  =  ( B `
 ( ( (
# `  ( A concat  B ) )  -  1 )  -  ( # `  A ) ) ) )
55 lsw 12388 . . . 4  |-  ( B  e. Word  V  ->  ( lastS  `  B )  =  ( B `  ( (
# `  B )  -  1 ) ) )
56553ad2ant2 1010 . . 3  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  ( lastS  `  B )  =  ( B `  ( (
# `  B )  -  1 ) ) )
5732, 54, 563eqtr4d 2505 . 2  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  if ( ( ( # `  ( A concat  B ) )  -  1 )  <  ( # `  A
) ,  ( A `
 ( ( # `  ( A concat  B ) )  -  1 ) ) ,  ( B `
 ( ( (
# `  ( A concat  B ) )  -  1 )  -  ( # `  A ) ) ) )  =  ( lastS  `  B
) )
583, 12, 573eqtrd 2499 1  |-  ( ( A  e. Word  V  /\  B  e. Word  V  /\  B  =/=  (/) )  ->  ( lastS  `  ( A concat  B ) )  =  ( lastS  `  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3748   ifcif 3902   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9395   RRcr 9396   1c1 9398    + caddc 9400    < clt 9533    - cmin 9710   NNcn 10437   NN0cn0 10694   ZZcz 10761   #chash 12224  Word cword 12343   lastS clsw 12344   concat cconcat 12345
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-lsw 12352  df-concat 12353
This theorem is referenced by:  lswccats1  12434  lswccats1fst  30433
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