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Theorem ccatval2 11746
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 11742 . . 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 977 . 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 2496 . . . 4  |-  ( x  =  I  ->  (
x  e.  ( 0..^ ( # `  S
) )  <->  I  e.  ( 0..^ ( # `  S
) ) ) )
4 fveq2 5728 . . . 4  |-  ( x  =  I  ->  ( S `  x )  =  ( S `  I ) )
5 oveq1 6088 . . . . 5  |-  ( x  =  I  ->  (
x  -  ( # `  S ) )  =  ( I  -  ( # `
 S ) ) )
65fveq2d 5732 . . . 4  |-  ( x  =  I  ->  ( T `  ( x  -  ( # `  S
) ) )  =  ( T `  (
I  -  ( # `  S ) ) ) )
73, 4, 6ifbieq12d 3761 . . 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 11167 . . . . . 6  |-  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  =  (/)
9 minel 3683 . . . . . 6  |-  ( ( I  e.  ( (
# `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) )  /\  ( ( 0..^ ( # `  S
) )  i^i  (
( # `  S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) )  =  (/) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
108, 9mpan2 653 . . . . 5  |-  ( I  e.  ( ( # `  S )..^ ( (
# `  S )  +  ( # `  T
) ) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
11103ad2ant3 980 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  -.  I  e.  (
0..^ ( # `  S
) ) )
12 iffalse 3746 . . . 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 2490 . 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 11734 . . . . . 6  |-  ( S  e. Word  B  ->  S  e.  Fin )
1615adantr 452 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  e.  Fin )
17 hashcl 11639 . . . . 5  |-  ( S  e.  Fin  ->  ( # `
 S )  e. 
NN0 )
18 fzoss1 11162 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
19 nn0uz 10520 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
2018, 19eleq2s 2528 . . . . 5  |-  ( (
# `  S )  e.  NN0  ->  ( ( # `
 S )..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
C_  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
2116, 17, 203syl 19 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
2221sseld 3347 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) )  ->  I  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) ) )
23223impia 1150 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( ( # `  S
)..^ ( ( # `  S )  +  (
# `  T )
) ) )  ->  I  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
24 fvex 5742 . . 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 5810 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   ifcif 3739    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   Fincfn 7109   0cc0 8990    + caddc 8993    - cmin 9291   NN0cn0 10221   ZZ>=cuz 10488  ..^cfzo 11135   #chash 11618  Word cword 11717   concat cconcat 11718
This theorem is referenced by:  ccatval3  11747  ccatlid  11748  ccatass  11750  ccatswrd  11773  revccat  11798  ccatsymb  28179  swrdccatin12lem3  28212  swrdccatin12  28214  cshwidx  28242  lstccats1fst  28263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724
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