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Theorem ccatfval 12288
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 2996 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2996 . 2  |-  ( T  e.  W  ->  T  e.  _V )
3 fveq2 5706 . . . . . 6  |-  ( s  =  S  ->  ( # `
 s )  =  ( # `  S
) )
4 fveq2 5706 . . . . . 6  |-  ( t  =  T  ->  ( # `
 t )  =  ( # `  T
) )
53, 4oveqan12d 6125 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( # `  s
)  +  ( # `  t ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
65oveq2d 6122 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  =  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) )
73oveq2d 6122 . . . . . . 7  |-  ( s  =  S  ->  (
0..^ ( # `  s
) )  =  ( 0..^ ( # `  S
) ) )
87eleq2d 2510 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
98adantr 465 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
10 fveq1 5705 . . . . . 6  |-  ( s  =  S  ->  (
s `  x )  =  ( S `  x ) )
1110adantr 465 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s `  x
)  =  ( S `
 x ) )
12 simpr 461 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
133oveq2d 6122 . . . . . . 7  |-  ( s  =  S  ->  (
x  -  ( # `  s ) )  =  ( x  -  ( # `
 S ) ) )
1413adantr 465 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  -  ( # `
 s ) )  =  ( x  -  ( # `  S ) ) )
1512, 14fveq12d 5712 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( t `  (
x  -  ( # `  s ) ) )  =  ( T `  ( x  -  ( # `
 S ) ) ) )
169, 11, 15ifbieq12d 3831 . . . 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 4385 . . 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 12246 . . 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 6131 . . . 4  |-  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) )  e.  _V
2019mptex 5963 . . 3  |-  ( x  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) ) )  e.  _V
2117, 18, 20ovmpt2a 6236 . 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 477 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 1369    e. wcel 1756   _Vcvv 2987   ifcif 3806    e. cmpt 4365   ` cfv 5433  (class class class)co 6106   0cc0 9297    + caddc 9300    - cmin 9610  ..^cfzo 11563   #chash 12118   concat cconcat 12238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-concat 12246
This theorem is referenced by:  ccatcl  12289  ccatlen  12290  ccatval1  12291  ccatval2  12292  ccatvalfn  12295  repswccat  12438  ccatco  12478  ccatmulgnn0dir  26955  ofccat  26956
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