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Theorem itg1val 19528
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 5054 . . . . 5  |-  ( f  =  F  ->  ran  f  =  ran  F )
21difeq1d 3424 . . . 4  |-  ( f  =  F  ->  ( ran  f  \  { 0 } )  =  ( ran  F  \  {
0 } ) )
3 cnveq 5005 . . . . . . . 8  |-  ( f  =  F  ->  `' f  =  `' F
)
43imaeq1d 5161 . . . . . . 7  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
54fveq2d 5691 . . . . . 6  |-  ( f  =  F  ->  ( vol `  ( `' f
" { x }
) )  =  ( vol `  ( `' F " { x } ) ) )
65oveq2d 6056 . . . . 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 12455 . . 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 19466 . . 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 12436 . . 3  |-  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) )  e.  _V
118, 9, 10fvmpt 5765 . 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 12436 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
1312, 9dmmpti 5533 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
1411, 13eleq2s 2496 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 2670    \ cdif 3277   {csn 3774   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   RRcr 8945   0cc0 8946    x. cmul 8951   sum_csu 12434   volcvol 19313  MblFncmbf 19459   S.1citg1 19460
This theorem is referenced by:  itg1val2  19529  itg1cl  19530  itg1ge0  19531  itg10  19533  itg11  19536  itg1addlem5  19545  itg1mulc  19549  itg10a  19555  itg1ge0a  19556  itg1climres  19559
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-sum 12435  df-itg1 19466
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