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Theorem itg1val 19532
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 5058 . . . . 5  |-  ( f  =  F  ->  ran  f  =  ran  F )
21difeq1d 3428 . . . 4  |-  ( f  =  F  ->  ( ran  f  \  { 0 } )  =  ( ran  F  \  {
0 } ) )
3 cnveq 5009 . . . . . . . 8  |-  ( f  =  F  ->  `' f  =  `' F
)
43imaeq1d 5165 . . . . . . 7  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
54fveq2d 5695 . . . . . 6  |-  ( f  =  F  ->  ( vol `  ( `' f
" { x }
) )  =  ( vol `  ( `' F " { x } ) ) )
65oveq2d 6060 . . . . 5  |-  ( f  =  F  ->  (
x  x.  ( vol `  ( `' f " { x } ) ) )  =  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
76adantr 452 . . . 4  |-  ( ( f  =  F  /\  x  e.  ( ran  f  \  { 0 } ) )  ->  (
x  x.  ( vol `  ( `' f " { x } ) ) )  =  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
82, 7sumeq12dv 12459 . . 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 19470 . . 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 12440 . . 3  |-  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) )  e.  _V
118, 9, 10fvmpt 5769 . 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 12440 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
1312, 9dmmpti 5537 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
1411, 13eleq2s 2500 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 936    = wceq 1649    e. wcel 1721   {crab 2674    \ cdif 3281   {csn 3778   `'ccnv 4840   dom cdm 4841   ran crn 4842   "cima 4844   -->wf 5413   ` cfv 5417  (class class class)co 6044   Fincfn 7072   RRcr 8949   0cc0 8950    x. cmul 8955   sum_csu 12438   volcvol 19317  MblFncmbf 19463   S.1citg1 19464
This theorem is referenced by:  itg1val2  19533  itg1cl  19534  itg1ge0  19535  itg10  19537  itg11  19540  itg1addlem5  19549  itg1mulc  19553  itg10a  19559  itg1ge0a  19560  itg1climres  19563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-seq 11283  df-sum 12439  df-itg1 19470
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