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Theorem itg1val 21166
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 5070 . . . . 5  |-  ( f  =  F  ->  ran  f  =  ran  F )
21difeq1d 3478 . . . 4  |-  ( f  =  F  ->  ( ran  f  \  { 0 } )  =  ( ran  F  \  {
0 } ) )
3 cnveq 5018 . . . . . . . 8  |-  ( f  =  F  ->  `' f  =  `' F
)
43imaeq1d 5173 . . . . . . 7  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
54fveq2d 5700 . . . . . 6  |-  ( f  =  F  ->  ( vol `  ( `' f
" { x }
) )  =  ( vol `  ( `' F " { x } ) ) )
65oveq2d 6112 . . . . 5  |-  ( f  =  F  ->  (
x  x.  ( vol `  ( `' f " { x } ) ) )  =  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
76adantr 465 . . . 4  |-  ( ( f  =  F  /\  x  e.  ( ran  f  \  { 0 } ) )  ->  (
x  x.  ( vol `  ( `' f " { x } ) ) )  =  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
82, 7sumeq12dv 13188 . . 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 21105 . . 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 13170 . . 3  |-  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) )  e.  _V
118, 9, 10fvmpt 5779 . 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 13170 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
1312, 9dmmpti 5545 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
1411, 13eleq2s 2535 1  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   {crab 2724    \ cdif 3330   {csn 3882   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096   Fincfn 7315   RRcr 9286   0cc0 9287    x. cmul 9292   sum_csu 13168   volcvol 20952  MblFncmbf 21099   S.1citg1 21100
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-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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-nel 2614  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-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-sum 13169  df-itg1 21105
This theorem is referenced by:  itg1val2  21167  itg1cl  21168  itg1ge0  21169  itg10  21171  itg11  21174  itg1addlem5  21183  itg1mulc  21187  itg10a  21193  itg1ge0a  21194  itg1climres  21197
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