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Theorem sitmval 26739
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 2986 . . 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 6103 . . . . 5  |-  ( w  =  W  ->  (
wsitg m )  =  ( Wsitg m ) )
65dmeqd 5047 . . . 4  |-  ( w  =  W  ->  dom  ( wsitg m )  =  dom  ( Wsitg m
) )
7 fveq2 5696 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
8 ofeq 6327 . . . . . . 7  |-  ( (
dist `  w )  =  ( dist `  W
)  ->  oF
( dist `  w )  =  oF ( dist `  W ) )
97, 8syl 16 . . . . . 6  |-  ( w  =  W  ->  oF ( dist `  w
)  =  oF ( dist `  W
) )
109oveqd 6113 . . . . 5  |-  ( w  =  W  ->  (
f  oF (
dist `  w )
g )  =  ( f  oF (
dist `  W )
g ) )
1110fveq2d 5700 . . . 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 6153 . . 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 6104 . . . . 5  |-  ( m  =  M  ->  ( Wsitg m )  =  ( Wsitg M ) )
1413dmeqd 5047 . . . 4  |-  ( m  =  M  ->  dom  ( Wsitg m )  =  dom  ( Wsitg M
) )
15 oveq2 6104 . . . . 5  |-  ( m  =  M  ->  (
( RR*ss  ( 0 [,] +oo ) )sitg m )  =  ( ( RR*ss  ( 0 [,] +oo ) )sitg M ) )
16 sitmval.d . . . . . . . . 9  |-  D  =  ( dist `  W
)
1716eqcomi 2447 . . . . . . . 8  |-  ( dist `  W )  =  D
18 ofeq 6327 . . . . . . . 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 6113 . . . . 5  |-  ( m  =  M  ->  (
f  oF (
dist `  W )
g )  =  ( f  oF D g ) )
2215, 21fveq12d 5702 . . . 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 6153 . . 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 26722 . . 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 6121 . . . . 5  |-  ( Wsitg M )  e.  _V
2625dmex 6516 . . . 4  |-  dom  ( Wsitg M )  e.  _V
2726, 26mpt2ex 6655 . . 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 6231 . 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 1369    e. wcel 1756   _Vcvv 2977   U.cuni 4096   dom cdm 4845   ran crn 4846   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    oFcof 6323   0cc0 9287   +oocpnf 9420   [,]cicc 11308   ↾s cress 14180   distcds 14252   RR*scxrs 14443  measurescmeas 26614  sitmcsitm 26719  sitgcsitg 26720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-1st 6582  df-2nd 6583  df-sitm 26722
This theorem is referenced by:  sitmfval  26740
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