MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccatval2 Structured version   Unicode version

Theorem ccatval2 12269
Description: Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Assertion
Ref Expression
ccatval2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  -> 
( ( S concat  T
) `  I )  =  ( T `  ( I  -  ( # `
 S ) ) ) )

Proof of Theorem ccatval2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ccatfval 12265 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S concat  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
213adant3 1008 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  -> 
( S concat  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
3 eleq1 2498 . . . 4  |-  ( x  =  I  ->  (
x  e.  ( 0..^ ( # `  S
) )  <->  I  e.  ( 0..^ ( # `  S
) ) ) )
4 fveq2 5686 . . . 4  |-  ( x  =  I  ->  ( S `  x )  =  ( S `  I ) )
5 oveq1 6093 . . . . 5  |-  ( x  =  I  ->  (
x  -  ( # `  S ) )  =  ( I  -  ( # `
 S ) ) )
65fveq2d 5690 . . . 4  |-  ( x  =  I  ->  ( T `  ( x  -  ( # `  S
) ) )  =  ( T `  (
I  -  ( # `  S ) ) ) )
73, 4, 6ifbieq12d 3811 . . 3  |-  ( x  =  I  ->  if ( x  e.  (
0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) )  =  if ( I  e.  ( 0..^ ( # `  S
) ) ,  ( S `  I ) ,  ( T `  ( I  -  ( # `
 S ) ) ) ) )
8 fzodisj 11575 . . . . . 6  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  =  (/)
9 minel 3729 . . . . . 6  |-  ( ( I  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) )  /\  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  =  (/) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
108, 9mpan2 671 . . . . 5  |-  ( I  e.  ( ( # `  S )..^ ( (
# `  S )  +  ( # `  T
) ) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
11103ad2ant3 1011 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
12 iffalse 3794 . . . 4  |-  ( -.  I  e.  ( 0..^ ( # `  S
) )  ->  if ( I  e.  (
0..^ ( # `  S
) ) ,  ( S `  I ) ,  ( T `  ( I  -  ( # `
 S ) ) ) )  =  ( T `  ( I  -  ( # `  S
) ) ) )
1311, 12syl 16 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  if ( I  e.  ( 0..^ ( # `  S
) ) ,  ( S `  I ) ,  ( T `  ( I  -  ( # `
 S ) ) ) )  =  ( T `  ( I  -  ( # `  S
) ) ) )
147, 13sylan9eqr 2492 . 2  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `
 S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  /\  x  =  I )  ->  if ( x  e.  (
0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) )  =  ( T `  ( I  -  ( # `  S
) ) ) )
15 wrdfin 12240 . . . . . 6  |-  ( S  e. Word  B  ->  S  e.  Fin )
1615adantr 465 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  e.  Fin )
17 hashcl 12118 . . . . 5  |-  ( S  e.  Fin  ->  ( # `
 S )  e. 
NN0 )
18 fzoss1 11568 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
19 nn0uz 10887 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
2018, 19eleq2s 2530 . . . . 5  |-  ( (
# `  S )  e.  NN0  ->  ( ( # `
 S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
C_  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
2116, 17, 203syl 20 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
2221sseld 3350 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  ->  I  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) ) )
23223impia 1184 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  I  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
24 fvex 5696 . . 3  |-  ( T `
 ( I  -  ( # `  S ) ) )  e.  _V
2524a1i 11 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  -> 
( T `  (
I  -  ( # `  S ) ) )  e.  _V )
262, 14, 23, 25fvmptd 5774 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  -> 
( ( S concat  T
) `  I )  =  ( T `  ( I  -  ( # `
 S ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322    C_ wss 3323   (/)c0 3632   ifcif 3786    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Fincfn 7302   0cc0 9274    + caddc 9277    - cmin 9587   NN0cn0 10571   ZZ>=cuz 10853  ..^cfzo 11540   #chash 12095  Word cword 12213   concat cconcat 12215
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-concat 12223
This theorem is referenced by:  ccatval3  12270  ccatsymb  12273  ccatlid  12276  ccatass  12278  ccats1val2  12299  ccatswrd  12342  swrdccatin12lem2  12372  swrdccatin12  12374  revccat  12398  cshwidxmod  12432  ccat2s1p2  30219  ccatw2s1p1  30222  ccatw2s1p2  30223
  Copyright terms: Public domain W3C validator