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Theorem swrd2lsw 12552
Description: Extract the last two single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
Assertion
Ref Expression
swrd2lsw  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( W substr  <. ( ( # `  W )  -  2 ) ,  ( # `  W ) >. )  =  <" ( W `
 ( ( # `  W )  -  2 ) ) ( lastS  `  W
) "> )

Proof of Theorem swrd2lsw
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  W  e. Word  V )
2 lencl 12249 . . . . 5  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
3 1z 10676 . . . . . . . . 9  |-  1  e.  ZZ
4 nn0z 10669 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ZZ )
5 zltp1le 10694 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  ( # `  W )  e.  ZZ )  -> 
( 1  <  ( # `
 W )  <->  ( 1  +  1 )  <_ 
( # `  W ) ) )
63, 4, 5sylancr 663 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( 1  <  ( # `  W
)  <->  ( 1  +  1 )  <_  ( # `
 W ) ) )
7 1p1e2 10435 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
87a1i 11 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( 1  +  1 )  =  2 )
98breq1d 4302 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( (
1  +  1 )  <_  ( # `  W
)  <->  2  <_  ( # `
 W ) ) )
109biimpd 207 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( (
1  +  1 )  <_  ( # `  W
)  ->  2  <_  (
# `  W )
) )
116, 10sylbid 215 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( 1  <  ( # `  W
)  ->  2  <_  (
# `  W )
) )
1211imp 429 . . . . . 6  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  2  <_  ( # `  W
) )
13 2nn0 10596 . . . . . . . . 9  |-  2  e.  NN0
1413jctl 541 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( 2  e.  NN0  /\  ( # `
 W )  e. 
NN0 ) )
1514adantr 465 . . . . . . 7  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  (
2  e.  NN0  /\  ( # `  W )  e.  NN0 ) )
16 nn0sub 10630 . . . . . . 7  |-  ( ( 2  e.  NN0  /\  ( # `  W )  e.  NN0 )  -> 
( 2  <_  ( # `
 W )  <->  ( ( # `
 W )  - 
2 )  e.  NN0 ) )
1715, 16syl 16 . . . . . 6  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  (
2  <_  ( # `  W
)  <->  ( ( # `  W )  -  2 )  e.  NN0 )
)
1812, 17mpbid 210 . . . . 5  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  (
( # `  W )  -  2 )  e. 
NN0 )
192, 18sylan 471 . . . 4  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  (
( # `  W )  -  2 )  e. 
NN0 )
20 0red 9387 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  ZZ  ->  0  e.  RR )
21 1red 9401 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  ZZ  ->  1  e.  RR )
22 zre 10650 . . . . . . . . . . . 12  |-  ( (
# `  W )  e.  ZZ  ->  ( # `  W
)  e.  RR )
2320, 21, 223jca 1168 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  ZZ  ->  ( 0  e.  RR  /\  1  e.  RR  /\  ( # `  W )  e.  RR ) )
24 0lt1 9862 . . . . . . . . . . 11  |-  0  <  1
25 lttr 9451 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  ( # `
 W )  e.  RR )  ->  (
( 0  <  1  /\  1  <  ( # `  W ) )  -> 
0  <  ( # `  W
) ) )
2625expd 436 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  ( # `
 W )  e.  RR )  ->  (
0  <  1  ->  ( 1  <  ( # `  W )  ->  0  <  ( # `  W
) ) ) )
2723, 24, 26mpisyl 18 . . . . . . . . . 10  |-  ( (
# `  W )  e.  ZZ  ->  ( 1  <  ( # `  W
)  ->  0  <  (
# `  W )
) )
28 elnnz 10656 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  ZZ  /\  0  <  ( # `  W ) ) )
2928simplbi2 625 . . . . . . . . . 10  |-  ( (
# `  W )  e.  ZZ  ->  ( 0  <  ( # `  W
)  ->  ( # `  W
)  e.  NN ) )
3027, 29syld 44 . . . . . . . . 9  |-  ( (
# `  W )  e.  ZZ  ->  ( 1  <  ( # `  W
)  ->  ( # `  W
)  e.  NN ) )
314, 30syl 16 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( 1  <  ( # `  W
)  ->  ( # `  W
)  e.  NN ) )
3231imp 429 . . . . . . 7  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  ( # `
 W )  e.  NN )
33 fzo0end 11619 . . . . . . 7  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  - 
1 )  e.  ( 0..^ ( # `  W
) ) )
3432, 33syl 16 . . . . . 6  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  (
( # `  W )  -  1 )  e.  ( 0..^ ( # `  W ) ) )
35 nn0cn 10589 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  CC )
36 2cn 10392 . . . . . . . . . . . 12  |-  2  e.  CC
3736a1i 11 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  2  e.  CC )
38 ax-1cn 9340 . . . . . . . . . . . 12  |-  1  e.  CC
3938a1i 11 . . . . . . . . . . 11  |-  ( (
# `  W )  e.  NN0  ->  1  e.  CC )
4035, 37, 393jca 1168 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  e.  CC  /\  2  e.  CC  /\  1  e.  CC ) )
41 1e2m1 10437 . . . . . . . . . . . . 13  |-  1  =  ( 2  -  1 )
4241a1i 11 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  1  =  ( 2  -  1 ) )
4342oveq2d 6107 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( # `  W )  -  1 )  =  ( ( # `  W
)  -  ( 2  -  1 ) ) )
44 subsub 9639 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( # `  W )  -  ( 2  -  1 ) )  =  ( ( ( # `  W )  -  2 )  +  1 ) )
4543, 44eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  CC  /\  2  e.  CC  /\  1  e.  CC )  ->  (
( # `  W )  -  1 )  =  ( ( ( # `  W )  -  2 )  +  1 ) )
4640, 45syl 16 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  - 
1 )  =  ( ( ( # `  W
)  -  2 )  +  1 ) )
4746eqcomd 2448 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  -  2 )  +  1 )  =  ( ( # `  W
)  -  1 ) )
4847eleq1d 2509 . . . . . . 7  |-  ( (
# `  W )  e.  NN0  ->  ( (
( ( # `  W
)  -  2 )  +  1 )  e.  ( 0..^ ( # `  W ) )  <->  ( ( # `
 W )  - 
1 )  e.  ( 0..^ ( # `  W
) ) ) )
4948adantr 465 . . . . . 6  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  (
( ( ( # `  W )  -  2 )  +  1 )  e.  ( 0..^ (
# `  W )
)  <->  ( ( # `  W )  -  1 )  e.  ( 0..^ ( # `  W
) ) ) )
5034, 49mpbird 232 . . . . 5  |-  ( ( ( # `  W
)  e.  NN0  /\  1  <  ( # `  W
) )  ->  (
( ( # `  W
)  -  2 )  +  1 )  e.  ( 0..^ ( # `  W ) ) )
512, 50sylan 471 . . . 4  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  (
( ( # `  W
)  -  2 )  +  1 )  e.  ( 0..^ ( # `  W ) ) )
521, 19, 513jca 1168 . . 3  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( W  e. Word  V  /\  (
( # `  W )  -  2 )  e. 
NN0  /\  ( (
( # `  W )  -  2 )  +  1 )  e.  ( 0..^ ( # `  W
) ) ) )
53 swrds2 12545 . . 3  |-  ( ( W  e. Word  V  /\  ( ( # `  W
)  -  2 )  e.  NN0  /\  (
( ( # `  W
)  -  2 )  +  1 )  e.  ( 0..^ ( # `  W ) ) )  ->  ( W substr  <. (
( # `  W )  -  2 ) ,  ( ( ( # `  W )  -  2 )  +  2 )
>. )  =  <" ( W `  (
( # `  W )  -  2 ) ) ( W `  (
( ( # `  W
)  -  2 )  +  1 ) ) "> )
5452, 53syl 16 . 2  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( W substr  <. ( ( # `  W )  -  2 ) ,  ( ( ( # `  W
)  -  2 )  +  2 ) >.
)  =  <" ( W `  ( ( # `
 W )  - 
2 ) ) ( W `  ( ( ( # `  W
)  -  2 )  +  1 ) ) "> )
5535, 36jctir 538 . . . . . 6  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  e.  CC  /\  2  e.  CC ) )
56 npcan 9619 . . . . . . 7  |-  ( ( ( # `  W
)  e.  CC  /\  2  e.  CC )  ->  ( ( ( # `  W )  -  2 )  +  2 )  =  ( # `  W
) )
5756eqcomd 2448 . . . . . 6  |-  ( ( ( # `  W
)  e.  CC  /\  2  e.  CC )  ->  ( # `  W
)  =  ( ( ( # `  W
)  -  2 )  +  2 ) )
582, 55, 573syl 20 . . . . 5  |-  ( W  e. Word  V  ->  ( # `
 W )  =  ( ( ( # `  W )  -  2 )  +  2 ) )
5958adantr 465 . . . 4  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( # `
 W )  =  ( ( ( # `  W )  -  2 )  +  2 ) )
6059opeq2d 4066 . . 3  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  <. (
( # `  W )  -  2 ) ,  ( # `  W
) >.  =  <. (
( # `  W )  -  2 ) ,  ( ( ( # `  W )  -  2 )  +  2 )
>. )
6160oveq2d 6107 . 2  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( W substr  <. ( ( # `  W )  -  2 ) ,  ( # `  W ) >. )  =  ( W substr  <. (
( # `  W )  -  2 ) ,  ( ( ( # `  W )  -  2 )  +  2 )
>. ) )
62 eqidd 2444 . . 3  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( W `  ( ( # `
 W )  - 
2 ) )  =  ( W `  (
( # `  W )  -  2 ) ) )
63 lsw 12266 . . . . 5  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( (
# `  W )  -  1 ) ) )
6440, 44syl 16 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  -  ( 2  -  1 ) )  =  ( ( ( # `  W
)  -  2 )  +  1 ) )
6564eqcomd 2448 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  -  2 )  +  1 )  =  ( ( # `  W
)  -  ( 2  -  1 ) ) )
66 2m1e1 10436 . . . . . . . . . . 11  |-  ( 2  -  1 )  =  1
6766a1i 11 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN0  ->  ( 2  -  1 )  =  1 )
6867oveq2d 6107 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN0  ->  ( ( # `
 W )  -  ( 2  -  1 ) )  =  ( ( # `  W
)  -  1 ) )
6965, 68eqtrd 2475 . . . . . . . 8  |-  ( (
# `  W )  e.  NN0  ->  ( (
( # `  W )  -  2 )  +  1 )  =  ( ( # `  W
)  -  1 ) )
702, 69syl 16 . . . . . . 7  |-  ( W  e. Word  V  ->  (
( ( # `  W
)  -  2 )  +  1 )  =  ( ( # `  W
)  -  1 ) )
7170eqcomd 2448 . . . . . 6  |-  ( W  e. Word  V  ->  (
( # `  W )  -  1 )  =  ( ( ( # `  W )  -  2 )  +  1 ) )
7271fveq2d 5695 . . . . 5  |-  ( W  e. Word  V  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =  ( W `  (
( ( # `  W
)  -  2 )  +  1 ) ) )
7363, 72eqtrd 2475 . . . 4  |-  ( W  e. Word  V  ->  ( lastS  `  W )  =  ( W `  ( ( ( # `  W
)  -  2 )  +  1 ) ) )
7473adantr 465 . . 3  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( lastS  `  W )  =  ( W `  ( ( ( # `  W
)  -  2 )  +  1 ) ) )
7562, 74s2eqd 12489 . 2  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  <" ( W `  ( ( # `
 W )  - 
2 ) ) ( lastS  `  W ) ">  =  <" ( W `
 ( ( # `  W )  -  2 ) ) ( W `
 ( ( (
# `  W )  -  2 )  +  1 ) ) "> )
7654, 61, 753eqtr4d 2485 1  |-  ( ( W  e. Word  V  /\  1  <  ( # `  W
) )  ->  ( W substr  <. ( ( # `  W )  -  2 ) ,  ( # `  W ) >. )  =  <" ( W `
 ( ( # `  W )  -  2 ) ) ( lastS  `  W
) "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cop 3883   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419    - cmin 9595   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646  ..^cfzo 11548   #chash 12103  Word cword 12221   lastS clsw 12222   substr csubstr 12225   <"cs2 12468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-concat 12231  df-s1 12232  df-substr 12233  df-s2 12475
This theorem is referenced by:  2swrd2eqwrdeq  12553
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