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Theorem ccatval2 12570
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 12566 . . 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 1015 . 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 2513 . . . 4  |-  ( x  =  I  ->  (
x  e.  ( 0..^ ( # `  S
) )  <->  I  e.  ( 0..^ ( # `  S
) ) ) )
4 fveq2 5852 . . . 4  |-  ( x  =  I  ->  ( S `  x )  =  ( S `  I ) )
5 oveq1 6284 . . . . 5  |-  ( x  =  I  ->  (
x  -  ( # `  S ) )  =  ( I  -  ( # `
 S ) ) )
65fveq2d 5856 . . . 4  |-  ( x  =  I  ->  ( T `  ( x  -  ( # `  S
) ) )  =  ( T `  (
I  -  ( # `  S ) ) ) )
73, 4, 6ifbieq12d 3949 . . 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 11833 . . . . . 6  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  =  (/)
9 minel 3864 . . . . . 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 1018 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
1211iffalsed 3933 . . 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
) ) ) )
137, 12sylan9eqr 2504 . 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
) ) ) )
14 wrdfin 12535 . . . . . 6  |-  ( S  e. Word  B  ->  S  e.  Fin )
1514adantr 465 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  e.  Fin )
16 hashcl 12402 . . . . 5  |-  ( S  e.  Fin  ->  ( # `
 S )  e. 
NN0 )
17 fzoss1 11826 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
18 nn0uz 11119 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
1917, 18eleq2s 2549 . . . . 5  |-  ( (
# `  S )  e.  NN0  ->  ( ( # `
 S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
C_  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
2015, 16, 193syl 20 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
2120sseld 3485 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  ->  I  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) ) )
22213impia 1192 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  I  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
23 fvex 5862 . . 3  |-  ( T `
 ( I  -  ( # `  S ) ) )  e.  _V
2423a1i 11 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  -> 
( T `  (
I  -  ( # `  S ) ) )  e.  _V )
252, 13, 22, 24fvmptd 5942 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 972    = wceq 1381    e. wcel 1802   _Vcvv 3093    i^i cin 3457    C_ wss 3458   (/)c0 3767   ifcif 3922    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   Fincfn 7514   0cc0 9490    + caddc 9493    - cmin 9805   NN0cn0 10796   ZZ>=cuz 11085  ..^cfzo 11798   #chash 12379  Word cword 12508   concat cconcat 12510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-hash 12380  df-word 12516  df-concat 12518
This theorem is referenced by:  ccatval3  12571  ccatsymb  12574  ccatlid  12577  ccatass  12579  ccatrn  12580  ccats1val2  12605  ccat2s1p2  12607  ccatw2s1p1  12614  ccatw2s1p2  12615  ccatswrd  12655  swrdccatin12lem2  12688  swrdccatin12  12690  revccat  12714  cshwidxmod  12748
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