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Theorem ccatval2 12548
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 12544 . . 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 1011 . 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 2532 . . . 4  |-  ( x  =  I  ->  (
x  e.  ( 0..^ ( # `  S
) )  <->  I  e.  ( 0..^ ( # `  S
) ) ) )
4 fveq2 5857 . . . 4  |-  ( x  =  I  ->  ( S `  x )  =  ( S `  I ) )
5 oveq1 6282 . . . . 5  |-  ( x  =  I  ->  (
x  -  ( # `  S ) )  =  ( I  -  ( # `
 S ) ) )
65fveq2d 5861 . . . 4  |-  ( x  =  I  ->  ( T `  ( x  -  ( # `  S
) ) )  =  ( T `  (
I  -  ( # `  S ) ) ) )
73, 4, 6ifbieq12d 3959 . . 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 11816 . . . . . 6  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  =  (/)
9 minel 3875 . . . . . 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 1014 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
12 iffalse 3941 . . . 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 2523 . 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 12514 . . . . . 6  |-  ( S  e. Word  B  ->  S  e.  Fin )
1615adantr 465 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  e.  Fin )
17 hashcl 12383 . . . . 5  |-  ( S  e.  Fin  ->  ( # `
 S )  e. 
NN0 )
18 fzoss1 11809 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
19 nn0uz 11105 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
2018, 19eleq2s 2568 . . . . 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 3496 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  ->  I  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) ) )
23223impia 1188 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  I  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
24 fvex 5867 . . 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 5946 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    i^i cin 3468    C_ wss 3469   (/)c0 3778   ifcif 3932    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481    + caddc 9484    - cmin 9794   NN0cn0 10784   ZZ>=cuz 11071  ..^cfzo 11781   #chash 12360  Word cword 12487   concat cconcat 12489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-concat 12497
This theorem is referenced by:  ccatval3  12549  ccatsymb  12552  ccatlid  12555  ccatass  12557  ccats1val2  12581  ccat2s1p2  12583  ccatw2s1p1  12590  ccatw2s1p2  12591  ccatswrd  12631  swrdccatin12lem2  12664  swrdccatin12  12666  revccat  12690  cshwidxmod  12724
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