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Theorem ccatfval 12551
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 ++  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
Distinct variable groups:    x, S    x, T
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem ccatfval
Dummy variables  t 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3065 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 3065 . 2  |-  ( T  e.  W  ->  T  e.  _V )
3 fveq2 5803 . . . . . 6  |-  ( s  =  S  ->  ( # `
 s )  =  ( # `  S
) )
4 fveq2 5803 . . . . . 6  |-  ( t  =  T  ->  ( # `
 t )  =  ( # `  T
) )
53, 4oveqan12d 6251 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( # `  s
)  +  ( # `  t ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
65oveq2d 6248 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  =  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) )
73oveq2d 6248 . . . . . . 7  |-  ( s  =  S  ->  (
0..^ ( # `  s
) )  =  ( 0..^ ( # `  S
) ) )
87eleq2d 2470 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
98adantr 463 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
10 fveq1 5802 . . . . . 6  |-  ( s  =  S  ->  (
s `  x )  =  ( S `  x ) )
1110adantr 463 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s `  x
)  =  ( S `
 x ) )
12 simpr 459 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
133oveq2d 6248 . . . . . . 7  |-  ( s  =  S  ->  (
x  -  ( # `  s ) )  =  ( x  -  ( # `
 S ) ) )
1413adantr 463 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  -  ( # `
 s ) )  =  ( x  -  ( # `  S ) ) )
1512, 14fveq12d 5809 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( t `  (
x  -  ( # `  s ) ) )  =  ( T `  ( x  -  ( # `
 S ) ) ) )
169, 11, 15ifbieq12d 3909 . . . 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 4470 . . 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 12498 . . 3  |- ++  =  ( s  e.  _V , 
t  e.  _V  |->  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) ) )
19 ovex 6260 . . . 4  |-  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) )  e.  _V
2019mptex 6078 . . 3  |-  ( x  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) ) )  e.  _V
2117, 18, 20ovmpt2a 6368 . 2  |-  ( ( S  e.  _V  /\  T  e.  _V )  ->  ( S ++  T )  =  ( x  e.  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) )  |->  if ( x  e.  ( 0..^ ( # `  S
) ) ,  ( S `  x ) ,  ( T `  ( x  -  ( # `
 S ) ) ) ) ) )
221, 2, 21syl2an 475 1  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ++  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 367    = wceq 1403    e. wcel 1840   _Vcvv 3056   ifcif 3882    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232   0cc0 9440    + caddc 9443    - cmin 9759  ..^cfzo 11765   #chash 12357   ++ cconcat 12490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-concat 12498
This theorem is referenced by:  ccatcl  12552  ccatlen  12553  ccatval1  12554  ccatval2  12555  ccatvalfn  12558  repswccat  12718  ccatco  12762  ccatmulgnn0dir  28883  ofccat  28884
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