Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ccat2s1fvw Structured version   Unicode version

Theorem ccat2s1fvw 30412
Description: A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.)
Assertion
Ref Expression
ccat2s1fvw  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
( ( W concat  <" X "> ) concat  <" Y "> ) `  I
)  =  ( W `
 I ) )

Proof of Theorem ccat2s1fvw
StepHypRef Expression
1 simp1 988 . . . . 5  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  W  e. Word  V )
21anim1i 568 . . . 4  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  ( W  e. Word  V  /\  ( X  e.  V  /\  Y  e.  V )
) )
3 3anass 969 . . . 4  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  <->  ( W  e. Word  V  /\  ( X  e.  V  /\  Y  e.  V
) ) )
42, 3sylibr 212 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V ) )
5 ccatw2s1ccatws2 12665 . . . 4  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W concat  <" X "> ) concat  <" Y "> )  =  ( W concat  <" X Y "> ) )
65fveq1d 5794 . . 3  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( ( W concat  <" X "> ) concat  <" Y "> ) `  I )  =  ( ( W concat  <" X Y "> ) `  I ) )
74, 6syl 16 . 2  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
( ( W concat  <" X "> ) concat  <" Y "> ) `  I
)  =  ( ( W concat  <" X Y "> ) `  I ) )
81adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  W  e. Word  V )
9 s2cl 12614 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  <" X Y ">  e. Word  V
)
109adantl 466 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  <" X Y ">  e. Word  V
)
11 simp2 989 . . . . 5  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  I  e.  NN0 )
12 lencl 12360 . . . . . . 7  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
1312nn0zd 10849 . . . . . 6  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  ZZ )
14133ad2ant1 1009 . . . . 5  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  ( # `
 W )  e.  ZZ )
15 simp3 990 . . . . 5  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  I  <  ( # `  W
) )
16 elfzo0z 11699 . . . . 5  |-  ( I  e.  ( 0..^ (
# `  W )
)  <->  ( I  e. 
NN0  /\  ( # `  W
)  e.  ZZ  /\  I  <  ( # `  W
) ) )
1711, 14, 15, 16syl3anbrc 1172 . . . 4  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  I  e.  ( 0..^ ( # `  W ) ) )
1817adantr 465 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  I  e.  ( 0..^ ( # `  W ) ) )
19 ccatval1 12387 . . 3  |-  ( ( W  e. Word  V  /\  <" X Y ">  e. Word  V  /\  I  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W concat  <" X Y "> ) `  I )  =  ( W `  I ) )
208, 10, 18, 19syl3anc 1219 . 2  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
( W concat  <" X Y "> ) `  I )  =  ( W `  I ) )
217, 20eqtrd 2492 1  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
( ( W concat  <" X "> ) concat  <" Y "> ) `  I
)  =  ( W `
 I ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   0cc0 9386    < clt 9522   NN0cn0 10683   ZZcz 10750  ..^cfzo 11658   #chash 12213  Word cword 12332   concat cconcat 12334   <"cs1 12335   <"cs2 12579
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 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
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 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-s2 12586
This theorem is referenced by:  ccat2s1fst  30413  numclwwlkovf2ex  30820
  Copyright terms: Public domain W3C validator