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Theorem ccatfval 11428
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 2796 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2796 . 2  |-  ( T  e.  W  ->  T  e.  _V )
3 fveq2 5525 . . . . . 6  |-  ( s  =  S  ->  ( # `
 s )  =  ( # `  S
) )
4 fveq2 5525 . . . . . 6  |-  ( t  =  T  ->  ( # `
 t )  =  ( # `  T
) )
53, 4oveqan12d 5877 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( # `  s
)  +  ( # `  t ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
65oveq2d 5874 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  =  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) )
73oveq2d 5874 . . . . . . 7  |-  ( s  =  S  ->  (
0..^ ( # `  s
) )  =  ( 0..^ ( # `  S
) ) )
87eleq2d 2350 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
98adantr 451 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
10 fveq1 5524 . . . . . 6  |-  ( s  =  S  ->  (
s `  x )  =  ( S `  x ) )
1110adantr 451 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s `  x
)  =  ( S `
 x ) )
12 simpr 447 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
133oveq2d 5874 . . . . . . 7  |-  ( s  =  S  ->  (
x  -  ( # `  s ) )  =  ( x  -  ( # `
 S ) ) )
1413adantr 451 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  -  ( # `
 s ) )  =  ( x  -  ( # `  S ) ) )
1512, 14fveq12d 5531 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( t `  (
x  -  ( # `  s ) ) )  =  ( T `  ( x  -  ( # `
 S ) ) ) )
169, 11, 15ifbieq12d 3587 . . . 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 4098 . . 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 11410 . . 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 5883 . . . 4  |-  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) )  e.  _V
2019mptex 5746 . . 3  |-  ( x  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) ) )  e.  _V
2117, 18, 20ovmpt2a 5978 . 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 463 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737    + caddc 8740    - cmin 9037  ..^cfzo 10870   #chash 11337   concat cconcat 11404
This theorem is referenced by:  ccatcl  11429  ccatlen  11430  ccatval1  11431  ccatval2  11432  ccatco  11490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-concat 11410
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