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Theorem pcofval 20695
Description: The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
Assertion
Ref Expression
pcofval  |-  ( *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 ) ) ) ) )
Distinct variable group:    f, g, x, J

Proof of Theorem pcofval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 oveq2 6195 . . . 4  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
2 eqidd 2452 . . . 4  |-  ( j  =  J  ->  (
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 ) ) ) ) )
31, 1, 2mpt2eq123dv 6244 . . 3  |-  ( j  =  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 ) ) ) ) )  =  ( 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 ) ) ) ) ) )
4 df-pco 20690 . . 3  |-  *p  =  ( j  e.  Top  |->  ( 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 ) ) ) ) ) )
5 ovex 6212 . . . 4  |-  ( II 
Cn  J )  e. 
_V
65, 5mpt2ex 6747 . . 3  |-  ( 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 ) ) ) ) )  e.  _V
73, 4, 6fvmpt 5870 . 2  |-  ( J  e.  Top  ->  ( *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 ) ) ) ) ) )
84dmmptss 5429 . . . . . 6  |-  dom  *p  C_ 
Top
98sseli 3447 . . . . 5  |-  ( J  e.  dom  *p  ->  J  e.  Top )
109con3i 135 . . . 4  |-  ( -.  J  e.  Top  ->  -.  J  e.  dom  *p )
11 ndmfv 5810 . . . 4  |-  ( -.  J  e.  dom  *p  ->  ( *p `  J
)  =  (/) )
1210, 11syl 16 . . 3  |-  ( -.  J  e.  Top  ->  ( *p `  J )  =  (/) )
13 cntop2 18958 . . . . . . 7  |-  ( f  e.  ( II  Cn  J )  ->  J  e.  Top )
1413con3i 135 . . . . . 6  |-  ( -.  J  e.  Top  ->  -.  f  e.  ( II 
Cn  J ) )
1514eq0rdv 3767 . . . . 5  |-  ( -.  J  e.  Top  ->  ( II  Cn  J )  =  (/) )
16 mpt2eq12 6242 . . . . 5  |-  ( ( ( II  Cn  J
)  =  (/)  /\  (
II  Cn  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 ) ) ) ) )  =  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( f `  (
2  x.  x ) ) ,  ( g `
 ( ( 2  x.  x )  - 
1 ) ) ) ) ) )
1715, 15, 16syl2anc 661 . . . 4  |-  ( -.  J  e.  Top  ->  ( 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 ) ) ) ) )  =  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( f `  (
2  x.  x ) ) ,  ( g `
 ( ( 2  x.  x )  - 
1 ) ) ) ) ) )
18 mpt20 6252 . . . 4  |-  ( f  e.  (/) ,  g  e.  (/)  |->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( f `  ( 2  x.  x ) ) ,  ( g `  ( ( 2  x.  x )  -  1 ) ) ) ) )  =  (/)
1917, 18syl6eq 2507 . . 3  |-  ( -.  J  e.  Top  ->  ( 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 ) ) ) ) )  =  (/) )
2012, 19eqtr4d 2494 . 2  |-  ( -.  J  e.  Top  ->  ( *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 ) ) ) ) ) )
217, 20pm2.61i 164 1  |-  ( *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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   (/)c0 3732   ifcif 3886   class class class wbr 4387    |-> cmpt 4445   dom cdm 4935   ` cfv 5513  (class class class)co 6187    |-> cmpt2 6189   0cc0 9380   1c1 9381    x. cmul 9385    <_ cle 9517    - cmin 9693    / cdiv 10091   2c2 10469   [,]cicc 11401   Topctop 18611    Cn ccn 18941   IIcii 20564   *pcpco 20685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-map 7313  df-top 18616  df-topon 18619  df-cn 18944  df-pco 20690
This theorem is referenced by:  pcoval  20696
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