MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccatfval Structured version   Unicode version

Theorem ccatfval 11744
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 2966 . 2  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2966 . 2  |-  ( T  e.  W  ->  T  e.  _V )
3 fveq2 5730 . . . . . 6  |-  ( s  =  S  ->  ( # `
 s )  =  ( # `  S
) )
4 fveq2 5730 . . . . . 6  |-  ( t  =  T  ->  ( # `
 t )  =  ( # `  T
) )
53, 4oveqan12d 6102 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( ( # `  s
)  +  ( # `  t ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
65oveq2d 6099 . . . 4  |-  ( ( s  =  S  /\  t  =  T )  ->  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  =  ( 0..^ ( (
# `  S )  +  ( # `  T
) ) ) )
73oveq2d 6099 . . . . . . 7  |-  ( s  =  S  ->  (
0..^ ( # `  s
) )  =  ( 0..^ ( # `  S
) ) )
87eleq2d 2505 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
98adantr 453 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  e.  ( 0..^ ( # `  s
) )  <->  x  e.  ( 0..^ ( # `  S
) ) ) )
10 fveq1 5729 . . . . . 6  |-  ( s  =  S  ->  (
s `  x )  =  ( S `  x ) )
1110adantr 453 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( s `  x
)  =  ( S `
 x ) )
12 simpr 449 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  t  =  T )
133oveq2d 6099 . . . . . . 7  |-  ( s  =  S  ->  (
x  -  ( # `  s ) )  =  ( x  -  ( # `
 S ) ) )
1413adantr 453 . . . . . 6  |-  ( ( s  =  S  /\  t  =  T )  ->  ( x  -  ( # `
 s ) )  =  ( x  -  ( # `  S ) ) )
1512, 14fveq12d 5736 . . . . 5  |-  ( ( s  =  S  /\  t  =  T )  ->  ( t `  (
x  -  ( # `  s ) ) )  =  ( T `  ( x  -  ( # `
 S ) ) ) )
169, 11, 15ifbieq12d 3763 . . . 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 4289 . . 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 11726 . . 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 6108 . . . 4  |-  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) )  e.  _V
2019mptex 5968 . . 3  |-  ( x  e.  ( 0..^ ( ( # `  S
)  +  ( # `  T ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `  x
) ,  ( T `
 ( x  -  ( # `  S ) ) ) ) )  e.  _V
2117, 18, 20ovmpt2a 6206 . 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 465 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   ifcif 3741    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   0cc0 8992    + caddc 8995    - cmin 9293  ..^cfzo 11137   #chash 11620   concat cconcat 11720
This theorem is referenced by:  ccatcl  11745  ccatlen  11746  ccatval1  11747  ccatval2  11748  ccatco  11806  ccatvalfn  28198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-concat 11726
  Copyright terms: Public domain W3C validator