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Theorem ccatfval 12269
Description: Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
ccatfval  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S concat  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
Distinct variable groups:    x, S    x, T    x, V    x, W

Proof of Theorem ccatfval
Dummy variables  t 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2979 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2979 . 2  |-  ( T  e.  W  ->  T  e.  _V )
3 fveq2 5688 . . . . . 6  |-  ( s  =  S  ->  ( # `
 s )  =  ( # `  S
) )
4 fveq2 5688 . . . . . 6  |-  ( t  =  T  ->  ( # `
 t )  =  ( # `  T
) )
53, 4oveqan12d 6109 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( # `  s
)  +  ( # `  t ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
65oveq2d 6106 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  =  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) )
73oveq2d 6106 . . . . . . 7  |-  ( s  =  S  ->  (
0..^ ( # `  s
) )  =  ( 0..^ ( # `  S
) ) )
87eleq2d 2508 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
98adantr 462 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
10 fveq1 5687 . . . . . 6  |-  ( s  =  S  ->  (
s `  x )  =  ( S `  x ) )
1110adantr 462 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s `  x
)  =  ( S `
 x ) )
12 simpr 458 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
133oveq2d 6106 . . . . . . 7  |-  ( s  =  S  ->  (
x  -  ( # `  s ) )  =  ( x  -  ( # `
 S ) ) )
1413adantr 462 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  -  ( # `
 s ) )  =  ( x  -  ( # `  S ) ) )
1512, 14fveq12d 5694 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( t `  (
x  -  ( # `  s ) ) )  =  ( T `  ( x  -  ( # `
 S ) ) ) )
169, 11, 15ifbieq12d 3813 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) )  =  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) ) )
176, 16mpteq12dv 4367 . . 3  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  e.  ( 0..^ ( ( # `  s )  +  (
# `  t )
) )  |->  if ( x  e.  ( 0..^ ( # `  s
) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
 s ) ) ) ) )  =  ( x  e.  ( 0..^ ( ( # `  S )  +  (
# `  T )
) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
18 df-concat 12227 . . 3  |- concat  =  ( s  e.  _V , 
t  e.  _V  |->  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) ) )
19 ovex 6115 . . . 4  |-  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) )  e.  _V
2019mptex 5945 . . 3  |-  ( x  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) ) )  e.  _V
2117, 18, 20ovmpt2a 6220 . 2  |-  ( ( S  e.  _V  /\  T  e.  _V )  ->  ( S concat  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
221, 2, 21syl2an 474 1  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S concat  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   ifcif 3788    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   0cc0 9278    + caddc 9281    - cmin 9591  ..^cfzo 11544   #chash 12099   concat cconcat 12219
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 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-concat 12227
This theorem is referenced by:  ccatcl  12270  ccatlen  12271  ccatval1  12272  ccatval2  12273  ccatvalfn  12276  repswccat  12419  ccatco  12459  ccatmulgnn0dir  26870  ofccat  26871
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