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Theorem sitmval 28796
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 3068 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
31, 2syl 17 . 2  |-  ( ph  ->  W  e.  _V )
4 sitmval.2 . 2  |-  ( ph  ->  M  e.  U. ran measures )
5 oveq1 6285 . . . . 5  |-  ( w  =  W  ->  (
wsitg m )  =  ( Wsitg m ) )
65dmeqd 5026 . . . 4  |-  ( w  =  W  ->  dom  ( wsitg m )  =  dom  ( Wsitg m
) )
7 fveq2 5849 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
8 ofeq 6523 . . . . . . 7  |-  ( (
dist `  w )  =  ( dist `  W
)  ->  oF
( dist `  w )  =  oF ( dist `  W ) )
97, 8syl 17 . . . . . 6  |-  ( w  =  W  ->  oF ( dist `  w
)  =  oF ( dist `  W
) )
109oveqd 6295 . . . . 5  |-  ( w  =  W  ->  (
f  oF (
dist `  w )
g )  =  ( f  oF (
dist `  W )
g ) )
1110fveq2d 5853 . . . 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 6340 . . 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 6286 . . . . 5  |-  ( m  =  M  ->  ( Wsitg m )  =  ( Wsitg M ) )
1413dmeqd 5026 . . . 4  |-  ( m  =  M  ->  dom  ( Wsitg m )  =  dom  ( Wsitg M
) )
15 oveq2 6286 . . . . 5  |-  ( m  =  M  ->  (
( RR*ss  ( 0 [,] +oo ) )sitg m )  =  ( ( RR*ss  ( 0 [,] +oo ) )sitg M ) )
16 sitmval.d . . . . . . . 8  |-  D  =  ( dist `  W
)
1716eqcomi 2415 . . . . . . 7  |-  ( dist `  W )  =  D
18 ofeq 6523 . . . . . . 7  |-  ( (
dist `  W )  =  D  ->  oF ( dist `  W
)  =  oF D )
1917, 18mp1i 13 . . . . . 6  |-  ( m  =  M  ->  oF ( dist `  W
)  =  oF D )
2019oveqd 6295 . . . . 5  |-  ( m  =  M  ->  (
f  oF (
dist `  W )
g )  =  ( f  oF D g ) )
2115, 20fveq12d 5855 . . . 4  |-  ( m  =  M  ->  (
( ( RR*ss  (
0 [,] +oo )
)sitg m ) `  ( f  oF ( dist `  W
) g ) )  =  ( ( (
RR*ss  ( 0 [,] +oo ) )sitg M ) `
 ( f  oF D g ) ) )
2214, 14, 21mpt2eq123dv 6340 . . 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 ) ) ) )
23 df-sitm 28779 . . 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 ) ) ) )
24 ovex 6306 . . . . 5  |-  ( Wsitg M )  e.  _V
2524dmex 6717 . . . 4  |-  dom  ( Wsitg M )  e.  _V
2625, 25mpt2ex 6861 . . 3  |-  ( f  e.  dom  ( Wsitg M ) ,  g  e.  dom  ( Wsitg M )  |->  ( ( ( RR*ss  ( 0 [,] +oo ) )sitg M ) `  (
f  oF D g ) ) )  e.  _V
2712, 22, 23, 26ovmpt2 6419 . 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 ) ) ) )
283, 4, 27syl2anc 659 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 1405    e. wcel 1842   _Vcvv 3059   U.cuni 4191   dom cdm 4823   ran crn 4824   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280    oFcof 6519   0cc0 9522   +oocpnf 9655   [,]cicc 11585   ↾s cress 14842   distcds 14918   RR*scxrs 15114  measurescmeas 28643  sitmcsitm 28776  sitgcsitg 28777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-1st 6784  df-2nd 6785  df-sitm 28779
This theorem is referenced by:  sitmfval  28797
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