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Theorem ccatw2s1p1 12597
Description: Extract the first symbol of a word concatenated with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
Assertion
Ref Expression
ccatw2s1p1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  N )  =  X )

Proof of Theorem ccatw2s1p1
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  W  e. Word  V )
21anim1i 568 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( W  e. Word  V  /\  ( X  e.  V  /\  Y  e.  V
) ) )
3 3anass 977 . . . . 5  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  <->  ( W  e. Word  V  /\  ( X  e.  V  /\  Y  e.  V
) ) )
42, 3sylibr 212 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V
) )
5 ccatw2s1ass 12591 . . . 4  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W concat  <" X "> ) concat  <" Y "> )  =  ( W concat  ( <" X "> concat  <" Y "> ) ) )
64, 5syl 16 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( W concat  <" X "> ) concat  <" Y "> )  =  ( W concat  ( <" X "> concat  <" Y "> ) ) )
76fveq1d 5866 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  N )  =  ( ( W concat 
( <" X "> concat 
<" Y "> ) ) `  N
) )
81adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  W  e. Word  V )
9 ccat2s1cl 12582 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( <" X "> concat  <" Y "> )  e. Word  V )
109adantl 466 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( <" X "> concat 
<" Y "> )  e. Word  V )
11 lencl 12522 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
12 eleq1 2539 . . . . . . . 8  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  e.  NN0  <->  N  e.  NN0 ) )
13 nn0z 10883 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
14 2z 10892 . . . . . . . . . . 11  |-  2  e.  ZZ
1514a1i 11 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  2  e.  ZZ )
1613, 15zaddcld 10966 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  2 )  e.  ZZ )
17 2pos 10623 . . . . . . . . . 10  |-  0  <  2
18 2re 10601 . . . . . . . . . . . 12  |-  2  e.  RR
1918a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  2  e.  RR )
20 nn0re 10800 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  N  e.  RR )
2119, 20ltaddposd 10132 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 0  <  2  <->  N  <  ( N  +  2 ) ) )
2217, 21mpbii 211 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  < 
( N  +  2 ) )
23 fzolb 11798 . . . . . . . . 9  |-  ( N  e.  ( N..^ ( N  +  2 ) )  <->  ( N  e.  ZZ  /\  ( N  +  2 )  e.  ZZ  /\  N  < 
( N  +  2 ) ) )
2413, 16, 22, 23syl3anbrc 1180 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ( N..^ ( N  +  2 ) ) )
2512, 24syl6bi 228 . . . . . . 7  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  e.  NN0  ->  N  e.  ( N..^ ( N  + 
2 ) ) ) )
2611, 25mpan9 469 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  N  e.  ( N..^ ( N  +  2
) ) )
27 id 22 . . . . . . . . 9  |-  ( (
# `  W )  =  N  ->  ( # `  W )  =  N )
28 oveq1 6289 . . . . . . . . 9  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )  +  2 )  =  ( N  +  2 ) )
2927, 28oveq12d 6300 . . . . . . . 8  |-  ( (
# `  W )  =  N  ->  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) )  =  ( N..^ ( N  +  2 ) ) )
3029eleq2d 2537 . . . . . . 7  |-  ( (
# `  W )  =  N  ->  ( N  e.  ( ( # `  W )..^ ( (
# `  W )  +  2 ) )  <-> 
N  e.  ( N..^ ( N  +  2 ) ) ) )
3130adantl 466 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( N  e.  ( ( # `  W
)..^ ( ( # `  W )  +  2 ) )  <->  N  e.  ( N..^ ( N  + 
2 ) ) ) )
3226, 31mpbird 232 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  ->  N  e.  ( ( # `
 W )..^ ( ( # `  W
)  +  2 ) ) )
3332adantr 465 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  e.  ( ( # `
 W )..^ ( ( # `  W
)  +  2 ) ) )
34 ccat2s1len 12585 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( # `  ( <" X "> concat  <" Y "> ) )  =  2 )
3534adantl 466 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( # `  ( <" X "> concat  <" Y "> ) )  =  2 )
3635oveq2d 6298 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( # `  W
)  +  ( # `  ( <" X "> concat  <" Y "> ) ) )  =  ( ( # `  W
)  +  2 ) )
3736oveq2d 6298 . . . 4  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( # `  W
)..^ ( ( # `  W )  +  (
# `  ( <" X "> concat  <" Y "> ) ) ) )  =  ( (
# `  W )..^ ( ( # `  W
)  +  2 ) ) )
3833, 37eleqtrrd 2558 . . 3  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  N  e.  ( ( # `
 W )..^ ( ( # `  W
)  +  ( # `  ( <" X "> concat  <" Y "> ) ) ) ) )
39 ccatval2 12555 . . 3  |-  ( ( W  e. Word  V  /\  ( <" X "> concat 
<" Y "> )  e. Word  V  /\  N  e.  ( ( # `  W
)..^ ( ( # `  W )  +  (
# `  ( <" X "> concat  <" Y "> ) ) ) ) )  ->  (
( W concat  ( <" X "> concat  <" Y "> ) ) `  N )  =  ( ( <" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) ) )
408, 10, 38, 39syl3anc 1228 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( W concat  ( <" X "> concat  <" Y "> ) ) `  N
)  =  ( (
<" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) ) )
41 oveq1 6289 . . . . . 6  |-  ( N  =  ( # `  W
)  ->  ( N  -  ( # `  W
) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
4241eqcoms 2479 . . . . 5  |-  ( (
# `  W )  =  N  ->  ( N  -  ( # `  W
) )  =  ( ( # `  W
)  -  ( # `  W ) ) )
4311nn0cnd 10850 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  CC )
4443subidd 9914 . . . . 5  |-  ( W  e. Word  V  ->  (
( # `  W )  -  ( # `  W
) )  =  0 )
4542, 44sylan9eqr 2530 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( N  -  ( # `
 W ) )  =  0 )
4645fveq2d 5868 . . 3  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  -> 
( ( <" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) )  =  ( ( <" X "> concat  <" Y "> ) `  0 ) )
47 ccat2s1p1 12589 . . 3  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( ( <" X "> concat  <" Y "> ) `  0 )  =  X )
4846, 47sylan9eq 2528 . 2  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( <" X "> concat  <" Y "> ) `  ( N  -  ( # `  W
) ) )  =  X )
497, 40, 483eqtrd 2512 1  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  N )  /\  ( X  e.  V  /\  Y  e.  V ) )  -> 
( ( ( W concat  <" X "> ) concat  <" Y "> ) `  N )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488    + caddc 9491    < clt 9624    - cmin 9801   2c2 10581   NN0cn0 10791   ZZcz 10860  ..^cfzo 11788   #chash 12367  Word cword 12494   concat cconcat 12496   <"cs1 12497
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505
This theorem is referenced by:  numclwwlkovf2ex  24760  numclwlk1lem2foa  24765
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