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

Theorem pcoval 21241
Description: The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
pcoval.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pcoval.3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
Assertion
Ref Expression
pcoval  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
Distinct variable groups:    x, F    x, G    ph, x    x, J

Proof of Theorem pcoval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcoval.2 . 2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 pcoval.3 . 2  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
3 fveq1 5858 . . . . . 6  |-  ( f  =  F  ->  (
f `  ( 2  x.  x ) )  =  ( F `  (
2  x.  x ) ) )
43adantr 465 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  (
2  x.  x ) )  =  ( F `
 ( 2  x.  x ) ) )
5 fveq1 5858 . . . . . 6  |-  ( g  =  G  ->  (
g `  ( (
2  x.  x )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
65adantl 466 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( g `  (
( 2  x.  x
)  -  1 ) )  =  ( G `
 ( ( 2  x.  x )  - 
1 ) ) )
74, 6ifeq12d 3954 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) )  =  if ( x  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  x
) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) )
87mpteq2dv 4529 . . 3  |-  ( ( f  =  F  /\  g  =  G )  ->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) )  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) ) ) )
9 pcofval 21240 . . 3  |-  ( *p
`  J )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
|->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( f `
 ( 2  x.  x ) ) ,  ( g `  (
( 2  x.  x
)  -  1 ) ) ) ) )
10 ovex 6302 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
1110mptex 6124 . . 3  |-  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )  e.  _V
128, 9, 11ovmpt2a 6410 . 2  |-  ( ( F  e.  ( II 
Cn  J )  /\  G  e.  ( II  Cn  J ) )  -> 
( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
131, 2, 12syl2anc 661 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   ifcif 3934   class class class wbr 4442    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484    x. cmul 9488    <_ cle 9620    - cmin 9796    / cdiv 10197   2c2 10576   [,]cicc 11523    Cn ccn 19486   IIcii 21109   *pcpco 21230
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-top 19161  df-topon 19164  df-cn 19489  df-pco 21235
This theorem is referenced by:  pcovalg  21242  pco1  21245  pcocn  21247  copco  21248  pcopt  21252  pcopt2  21253  pcoass  21254
  Copyright terms: Public domain W3C validator