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Theorem ccatval1 12547
Description: Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Assertion
Ref Expression
ccatval1  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  S ) ) )  ->  ( ( S concat  T ) `  I
)  =  ( S `
 I ) )

Proof of Theorem ccatval1
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.  ( 0..^ ( # `  S ) ) )  ->  ( 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 iftrue 3938 . . . 4  |-  ( I  e.  ( 0..^ (
# `  S )
)  ->  if (
I  e.  ( 0..^ ( # `  S
) ) ,  ( S `  I ) ,  ( T `  ( I  -  ( # `
 S ) ) ) )  =  ( S `  I ) )
983ad2ant3 1014 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  S ) ) )  ->  if ( I  e.  ( 0..^ (
# `  S )
) ,  ( S `
 I ) ,  ( T `  (
I  -  ( # `  S ) ) ) )  =  ( S `
 I ) )
107, 9sylan9eqr 2523 . 2  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  S
) ) )  /\  x  =  I )  ->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) )  =  ( S `  I
) )
11 wrdfin 12514 . . . . . . . . 9  |-  ( S  e. Word  B  ->  S  e.  Fin )
1211adantr 465 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  e.  Fin )
13 hashcl 12383 . . . . . . . 8  |-  ( S  e.  Fin  ->  ( # `
 S )  e. 
NN0 )
1412, 13syl 16 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  NN0 )
1514nn0zd 10953 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ZZ )
16 wrdfin 12514 . . . . . . . . . 10  |-  ( T  e. Word  B  ->  T  e.  Fin )
1716adantl 466 . . . . . . . . 9  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  T  e.  Fin )
18 hashcl 12383 . . . . . . . . 9  |-  ( T  e.  Fin  ->  ( # `
 T )  e. 
NN0 )
1917, 18syl 16 . . . . . . . 8  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  NN0 )
2019nn0zd 10953 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  ZZ )
2115, 20zaddcld 10959 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ZZ )
2214nn0red 10842 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  RR )
23 nn0addge1 10831 . . . . . . 7  |-  ( ( ( # `  S
)  e.  RR  /\  ( # `  T )  e.  NN0 )  -> 
( # `  S )  <_  ( ( # `  S )  +  (
# `  T )
) )
2422, 19, 23syl2anc 661 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  <_  ( ( # `
 S )  +  ( # `  T
) ) )
25 eluz2 11077 . . . . . 6  |-  ( ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) )  <->  ( ( # `
 S )  e.  ZZ  /\  ( (
# `  S )  +  ( # `  T
) )  e.  ZZ  /\  ( # `  S
)  <_  ( ( # `
 S )  +  ( # `  T
) ) ) )
2615, 21, 24, 25syl3anbrc 1175 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
27 fzoss2 11810 . . . . 5  |-  ( ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) )  ->  (
0..^ ( # `  S
) )  C_  (
0..^ ( ( # `  S )  +  (
# `  T )
) ) )
2826, 27syl 16 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( # `  S ) )  C_  ( 0..^ ( ( # `  S )  +  (
# `  T )
) ) )
2928sselda 3497 . . 3  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  I  e.  ( 0..^ ( # `  S
) ) )  ->  I  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) )
30293impa 1186 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  S ) ) )  ->  I  e.  ( 0..^ ( ( # `  S )  +  (
# `  T )
) ) )
31 fvex 5867 . . 3  |-  ( S `
 I )  e. 
_V
3231a1i 11 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  S ) ) )  ->  ( S `  I )  e.  _V )
332, 10, 30, 32fvmptd 5946 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  I  e.  ( 0..^ ( # `  S ) ) )  ->  ( ( S concat  T ) `  I
)  =  ( S `
 I ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   ifcif 3932   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Fincfn 7506   RRcr 9480   0cc0 9481    + caddc 9484    <_ cle 9618    - cmin 9794   NN0cn0 10784   ZZcz 10853   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:  ccatsymb  12552  ccatfv0  12553  ccatval1lsw  12554  ccatrid  12556  ccatass  12557  ccats1val1  12580  ccat2s1p1  12582  ccat2s1fvw  12592  ccatswrd  12631  swrdccat1  12632  cats1un  12651  swrdccatin1  12658  swrdccatin12lem3  12665  swrdccatin12  12666  splfv1  12681  splfv2a  12682  revccat  12690  cshwidxmod  12724  cats1fv  12774  ccat2s1fvwALT  12843  gsumccat  15825  efgsp1  16544  efgredlemd  16551  efgrelexlemb  16557  clwwlkel  24455  wwlkext2clwwlk  24465  signstfvn  28152  signstfvp  28154  signstfvneq0  28155
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