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Theorem itg1val 19253
Description: The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
itg1val  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Distinct variable group:    x, F

Proof of Theorem itg1val
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rneq 5007 . . . . 5  |-  ( f  =  F  ->  ran  f  =  ran  F )
21difeq1d 3380 . . . 4  |-  ( f  =  F  ->  ( ran  f  \  { 0 } )  =  ( ran  F  \  {
0 } ) )
3 cnveq 4958 . . . . . . . 8  |-  ( f  =  F  ->  `' f  =  `' F
)
43imaeq1d 5114 . . . . . . 7  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
54fveq2d 5636 . . . . . 6  |-  ( f  =  F  ->  ( vol `  ( `' f
" { x }
) )  =  ( vol `  ( `' F " { x } ) ) )
65oveq2d 5997 . . . . 5  |-  ( f  =  F  ->  (
x  x.  ( vol `  ( `' f " { x } ) ) )  =  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
76adantr 451 . . . 4  |-  ( ( f  =  F  /\  x  e.  ( ran  f  \  { 0 } ) )  ->  (
x  x.  ( vol `  ( `' f " { x } ) ) )  =  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
82, 7sumeq12dv 12387 . . 3  |-  ( f  =  F  ->  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  = 
sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
9 df-itg1 19191 . . 3  |-  S.1  =  ( f  e.  {
g  e. MblFn  |  (
g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR ) }  |->  sum_
x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f " {
x } ) ) ) )
10 sumex 12368 . . 3  |-  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) )  e.  _V
118, 9, 10fvmpt 5709 . 2  |-  ( F  e.  { g  e. MblFn  |  ( g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  e.  RR ) }  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
12 sumex 12368 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
1312, 9dmmpti 5478 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
1411, 13eleq2s 2458 1  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 935    = wceq 1647    e. wcel 1715   {crab 2632    \ cdif 3235   {csn 3729   `'ccnv 4791   dom cdm 4792   ran crn 4793   "cima 4795   -->wf 5354   ` cfv 5358  (class class class)co 5981   Fincfn 7006   RRcr 8883   0cc0 8884    x. cmul 8889   sum_csu 12366   volcvol 19038  MblFncmbf 19184   S.1citg1 19185
This theorem is referenced by:  itg1val2  19254  itg1cl  19255  itg1ge0  19256  itg10  19258  itg11  19261  itg1addlem5  19270  itg1mulc  19274  itg10a  19280  itg1ge0a  19281  itg1climres  19284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-seq 11211  df-sum 12367  df-itg1 19191
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