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Theorem lswn0 32607
Description: The last symbol of a not empty word exists. The empty set must be excluded as symbol, because otherwise, it cannot be distinguished between valid cases ( (/) is the last symbol) and invalid cases ( (/) means that no last symbol exists. This is because of the special definition of a function in set.mm. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
Assertion
Ref Expression
lswn0  |-  ( ( W  e. Word  V  /\  (/) 
e/  V  /\  ( # `
 W )  =/=  0 )  ->  ( lastS  `  W )  =/=  (/) )

Proof of Theorem lswn0
StepHypRef Expression
1 lsw 12573 . . 3  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
213ad2ant1 1015 . 2  |-  ( ( W  e. Word  V  /\  (/) 
e/  V  /\  ( # `
 W )  =/=  0 )  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
3 wrdf 12538 . . . . . 6  |-  ( W  e. Word  V  ->  W : ( 0..^ (
# `  W )
) --> V )
4 lencl 12549 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
5 simpll 751 . . . . . . . 8  |-  ( ( ( W : ( 0..^ ( # `  W
) ) --> V  /\  ( # `  W )  e.  NN0 )  /\  ( # `  W )  =/=  0 )  ->  W : ( 0..^ (
# `  W )
) --> V )
6 elnnne0 10805 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  NN0  /\  ( # `  W
)  =/=  0 ) )
76biimpri 206 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
( # `  W )  e.  NN )
8 nnm1nn0 10833 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  NN0 )
97, 8syl 16 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
( ( # `  W
)  -  1 )  e.  NN0 )
10 nn0re 10800 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  RR )
1110ltm1d 10473 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  - 
1 )  <  ( # `
 W ) )
1211adantr 463 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
( ( # `  W
)  -  1 )  <  ( # `  W
) )
13 elfzo0 11840 . . . . . . . . . 10  |-  ( ( ( # `  W
)  -  1 )  e.  ( 0..^ (
# `  W )
)  <->  ( ( (
# `  W )  -  1 )  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  ( ( # `  W
)  -  1 )  <  ( # `  W
) ) )
149, 7, 12, 13syl3anbrc 1178 . . . . . . . . 9  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
( ( # `  W
)  -  1 )  e.  ( 0..^ (
# `  W )
) )
1514adantll 711 . . . . . . . 8  |-  ( ( ( W : ( 0..^ ( # `  W
) ) --> V  /\  ( # `  W )  e.  NN0 )  /\  ( # `  W )  =/=  0 )  -> 
( ( # `  W
)  -  1 )  e.  ( 0..^ (
# `  W )
) )
165, 15ffvelrnd 6008 . . . . . . 7  |-  ( ( ( W : ( 0..^ ( # `  W
) ) --> V  /\  ( # `  W )  e.  NN0 )  /\  ( # `  W )  =/=  0 )  -> 
( W `  (
( # `  W )  -  1 ) )  e.  V )
1716ex 432 . . . . . 6  |-  ( ( W : ( 0..^ ( # `  W
) ) --> V  /\  ( # `  W )  e.  NN0 )  -> 
( ( # `  W
)  =/=  0  -> 
( W `  (
( # `  W )  -  1 ) )  e.  V ) )
183, 4, 17syl2anc 659 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  =/=  0  ->  ( W `  ( ( # `
 W )  - 
1 ) )  e.  V ) )
19 eleq1a 2537 . . . . . . . . . 10  |-  ( ( W `  ( (
# `  W )  -  1 ) )  e.  V  ->  ( (/)  =  ( W `  ( ( # `  W
)  -  1 ) )  ->  (/)  e.  V
) )
2019com12 31 . . . . . . . . 9  |-  ( (/)  =  ( W `  ( ( # `  W
)  -  1 ) )  ->  ( ( W `  ( ( # `
 W )  - 
1 ) )  e.  V  ->  (/)  e.  V
) )
2120eqcoms 2466 . . . . . . . 8  |-  ( ( W `  ( (
# `  W )  -  1 ) )  =  (/)  ->  ( ( W `  ( (
# `  W )  -  1 ) )  e.  V  ->  (/)  e.  V
) )
2221com12 31 . . . . . . 7  |-  ( ( W `  ( (
# `  W )  -  1 ) )  e.  V  ->  (
( W `  (
( # `  W )  -  1 ) )  =  (/)  ->  (/)  e.  V
) )
23 nnel 2799 . . . . . . 7  |-  ( -.  (/)  e/  V  <->  (/)  e.  V
)
2422, 23syl6ibr 227 . . . . . 6  |-  ( ( W `  ( (
# `  W )  -  1 ) )  e.  V  ->  (
( W `  (
( # `  W )  -  1 ) )  =  (/)  ->  -.  (/)  e/  V
) )
2524necon2ad 2667 . . . . 5  |-  ( ( W `  ( (
# `  W )  -  1 ) )  e.  V  ->  ( (/) 
e/  V  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =/=  (/) ) )
2618, 25syl6 33 . . . 4  |-  ( W  e. Word  V  ->  (
( # `  W )  =/=  0  ->  ( (/) 
e/  V  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =/=  (/) ) ) )
2726com23 78 . . 3  |-  ( W  e. Word  V  ->  ( (/) 
e/  V  ->  (
( # `  W )  =/=  0  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =/=  (/) ) ) )
28273imp 1188 . 2  |-  ( ( W  e. Word  V  /\  (/) 
e/  V  /\  ( # `
 W )  =/=  0 )  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =/=  (/) )
292, 28eqnetrd 2747 1  |-  ( ( W  e. Word  V  /\  (/) 
e/  V  /\  ( # `
 W )  =/=  0 )  ->  ( lastS  `  W )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    e/ wnel 2650   (/)c0 3783   class class class wbr 4439   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    < clt 9617    - cmin 9796   NNcn 10531   NN0cn0 10791  ..^cfzo 11799   #chash 12387  Word cword 12518   lastS clsw 12519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-lsw 12527
This theorem is referenced by: (None)
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