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Theorem sitmval 27927
Description: Value of the simple function integral metric for a given space  W and measure  M. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
sitmval.d  |-  D  =  ( dist `  W
)
sitmval.1  |-  ( ph  ->  W  e.  V )
sitmval.2  |-  ( ph  ->  M  e.  U. ran measures )
Assertion
Ref Expression
sitmval  |-  ( ph  ->  ( Wsitm M )  =  ( f  e. 
dom  ( Wsitg M
) ,  g  e. 
dom  ( Wsitg M
)  |->  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg M ) `
 ( f  oF D g ) ) ) )
Distinct variable groups:    f, g, M    f, W, g
Allowed substitution hints:    ph( f, g)    D( f, g)    V( f, g)

Proof of Theorem sitmval
Dummy variables  w  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitmval.1 . . 3  |-  ( ph  ->  W  e.  V )
2 elex 3122 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
31, 2syl 16 . 2  |-  ( ph  ->  W  e.  _V )
4 sitmval.2 . 2  |-  ( ph  ->  M  e.  U. ran measures )
5 oveq1 6289 . . . . 5  |-  ( w  =  W  ->  (
wsitg m )  =  ( Wsitg m ) )
65dmeqd 5203 . . . 4  |-  ( w  =  W  ->  dom  ( wsitg m )  =  dom  ( Wsitg m
) )
7 fveq2 5864 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
8 ofeq 6524 . . . . . . 7  |-  ( (
dist `  w )  =  ( dist `  W
)  ->  oF
( dist `  w )  =  oF ( dist `  W ) )
97, 8syl 16 . . . . . 6  |-  ( w  =  W  ->  oF ( dist `  w
)  =  oF ( dist `  W
) )
109oveqd 6299 . . . . 5  |-  ( w  =  W  ->  (
f  oF (
dist `  w )
g )  =  ( f  oF (
dist `  W )
g ) )
1110fveq2d 5868 . . . 4  |-  ( w  =  W  ->  (
( ( RR*ss  (
0 [,] +oo )
)sitg m ) `  ( f  oF ( dist `  w
) g ) )  =  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg m ) `
 ( f  oF ( dist `  W
) g ) ) )
126, 6, 11mpt2eq123dv 6341 . . 3  |-  ( w  =  W  ->  (
f  e.  dom  (
wsitg m ) ,  g  e.  dom  (
wsitg m )  |->  ( ( ( RR*ss  (
0 [,] +oo )
)sitg m ) `  ( f  oF ( dist `  w
) g ) ) )  =  ( f  e.  dom  ( Wsitg m ) ,  g  e.  dom  ( Wsitg m )  |->  ( ( ( RR*ss  ( 0 [,] +oo ) )sitg m ) `  (
f  oF (
dist `  W )
g ) ) ) )
13 oveq2 6290 . . . . 5  |-  ( m  =  M  ->  ( Wsitg m )  =  ( Wsitg M ) )
1413dmeqd 5203 . . . 4  |-  ( m  =  M  ->  dom  ( Wsitg m )  =  dom  ( Wsitg M
) )
15 oveq2 6290 . . . . 5  |-  ( m  =  M  ->  (
( RR*ss  ( 0 [,] +oo ) )sitg m )  =  ( ( RR*ss  ( 0 [,] +oo ) )sitg M ) )
16 sitmval.d . . . . . . . . 9  |-  D  =  ( dist `  W
)
1716eqcomi 2480 . . . . . . . 8  |-  ( dist `  W )  =  D
18 ofeq 6524 . . . . . . . 8  |-  ( (
dist `  W )  =  D  ->  oF ( dist `  W
)  =  oF D )
1917, 18ax-mp 5 . . . . . . 7  |-  oF ( dist `  W
)  =  oF D
2019a1i 11 . . . . . 6  |-  ( m  =  M  ->  oF ( dist `  W
)  =  oF D )
2120oveqd 6299 . . . . 5  |-  ( m  =  M  ->  (
f  oF (
dist `  W )
g )  =  ( f  oF D g ) )
2215, 21fveq12d 5870 . . . 4  |-  ( m  =  M  ->  (
( ( RR*ss  (
0 [,] +oo )
)sitg m ) `  ( f  oF ( dist `  W
) g ) )  =  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg M ) `
 ( f  oF D g ) ) )
2314, 14, 22mpt2eq123dv 6341 . . 3  |-  ( m  =  M  ->  (
f  e.  dom  ( Wsitg m ) ,  g  e.  dom  ( Wsitg m )  |->  ( ( ( RR*ss  ( 0 [,] +oo ) )sitg m ) `  (
f  oF (
dist `  W )
g ) ) )  =  ( f  e. 
dom  ( Wsitg M
) ,  g  e. 
dom  ( Wsitg M
)  |->  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg M ) `
 ( f  oF D g ) ) ) )
24 df-sitm 27910 . . 3  |- sitm  =  ( w  e.  _V ,  m  e.  U. ran measures  |->  ( f  e.  dom  ( wsitg m ) ,  g  e.  dom  ( wsitg m )  |->  ( ( ( RR*ss  ( 0 [,] +oo ) )sitg m ) `  (
f  oF (
dist `  w )
g ) ) ) )
25 ovex 6307 . . . . 5  |-  ( Wsitg M )  e.  _V
2625dmex 6714 . . . 4  |-  dom  ( Wsitg M )  e.  _V
2726, 26mpt2ex 6857 . . 3  |-  ( f  e.  dom  ( Wsitg M ) ,  g  e.  dom  ( Wsitg M )  |->  ( ( ( RR*ss  ( 0 [,] +oo ) )sitg M ) `  (
f  oF D g ) ) )  e.  _V
2812, 23, 24, 27ovmpt2 6420 . 2  |-  ( ( W  e.  _V  /\  M  e.  U. ran measures )  -> 
( Wsitm M )  =  ( f  e. 
dom  ( Wsitg M
) ,  g  e. 
dom  ( Wsitg M
)  |->  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg M ) `
 ( f  oF D g ) ) ) )
293, 4, 28syl2anc 661 1  |-  ( ph  ->  ( Wsitm M )  =  ( f  e. 
dom  ( Wsitg M
) ,  g  e. 
dom  ( Wsitg M
)  |->  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg M ) `
 ( f  oF D g ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   U.cuni 4245   dom cdm 4999   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    oFcof 6520   0cc0 9488   +oocpnf 9621   [,]cicc 11528   ↾s cress 14484   distcds 14557   RR*scxrs 14748  measurescmeas 27803  sitmcsitm 27907  sitgcsitg 27908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-1st 6781  df-2nd 6782  df-sitm 27910
This theorem is referenced by:  sitmfval  27928
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