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Theorem itg1val2 19039
Description: The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
itg1val2  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( S.1 `  F )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem itg1val2
StepHypRef Expression
1 itg1val 19038 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
21adantr 451 . 2  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
3 simpr2 962 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( ran  F 
\  { 0 } )  C_  A )
43sselda 3180 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( ran  F  \  { 0 } ) )  ->  x  e.  A )
5 simpr3 963 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
65sselda 3180 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  x  e.  ( RR  \  {
0 } ) )
7 eldifi 3298 . . . . . . 7  |-  ( x  e.  ( RR  \  { 0 } )  ->  x  e.  RR )
86, 7syl 15 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  x  e.  RR )
9 i1fima2sn 19035 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
109adantlr 695 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( RR  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
116, 10syldan 456 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
128, 11remulcld 8863 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  (
x  x.  ( vol `  ( `' F " { x } ) ) )  e.  RR )
1312recnd 8861 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  A )  ->  (
x  x.  ( vol `  ( `' F " { x } ) ) )  e.  CC )
144, 13syldan 456 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( ran  F  \  { 0 } ) )  ->  ( x  x.  ( vol `  ( `' F " { x } ) ) )  e.  CC )
15 i1ff 19031 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
1615ad2antrr 706 . . . . . . . . 9  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  F : RR --> RR )
17 ffn 5389 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
18 dffn3 5396 . . . . . . . . . 10  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
1917, 18sylib 188 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F : RR --> ran  F
)
2016, 19syl 15 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  F : RR --> ran  F )
21 eldifn 3299 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
2221adantl 452 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
23 difss 3303 . . . . . . . . . . . . . . 15  |-  ( A 
\  ( ran  F  \  { 0 } ) )  C_  A
24 simplr3 999 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
2523, 24syl5ss 3190 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( A  \ 
( ran  F  \  {
0 } ) ) 
C_  ( RR  \  { 0 } ) )
26 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  ( A  \  ( ran 
F  \  { 0 } ) ) )
2725, 26sseldd 3181 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  ( RR  \  { 0 } ) )
28 eldifn 3299 . . . . . . . . . . . . 13  |-  ( x  e.  ( RR  \  { 0 } )  ->  -.  x  e.  { 0 } )
2927, 28syl 15 . . . . . . . . . . . 12  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  -.  x  e.  { 0 } )
3029biantrud 493 . . . . . . . . . . 11  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  e. 
ran  F  <->  ( x  e. 
ran  F  /\  -.  x  e.  { 0 } ) ) )
31 eldif 3162 . . . . . . . . . . 11  |-  ( x  e.  ( ran  F  \  { 0 } )  <-> 
( x  e.  ran  F  /\  -.  x  e. 
{ 0 } ) )
3230, 31syl6rbbr 255 . . . . . . . . . 10  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  e.  ( ran  F  \  { 0 } )  <-> 
x  e.  ran  F
) )
3322, 32mtbid 291 . . . . . . . . 9  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  -.  x  e.  ran  F )
34 disjsn 3693 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  F )
3533, 34sylibr 203 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( ran  F  i^i  { x } )  =  (/) )
36 fimacnvdisj 5419 . . . . . . . 8  |-  ( ( F : RR --> ran  F  /\  ( ran  F  i^i  { x } )  =  (/) )  ->  ( `' F " { x } )  =  (/) )
3720, 35, 36syl2anc 642 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( `' F " { x } )  =  (/) )
3837fveq2d 5529 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( vol `  ( `' F " { x } ) )  =  ( vol `  (/) ) )
39 0mbl 18897 . . . . . . . 8  |-  (/)  e.  dom  vol
40 mblvol 18889 . . . . . . . 8  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
4139, 40ax-mp 8 . . . . . . 7  |-  ( vol `  (/) )  =  ( vol * `  (/) )
42 ovol0 18852 . . . . . . 7  |-  ( vol
* `  (/) )  =  0
4341, 42eqtri 2303 . . . . . 6  |-  ( vol `  (/) )  =  0
4438, 43syl6eq 2331 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( vol `  ( `' F " { x } ) )  =  0 )
4544oveq2d 5874 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  x.  ( vol `  ( `' F " { x } ) ) )  =  ( x  x.  0 ) )
46 eldifi 3298 . . . . . . 7  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  x  e.  A
)
4746, 8sylan2 460 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  RR )
4847recnd 8861 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  x  e.  CC )
4948mul01d 9011 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  x.  0 )  =  0 )
5045, 49eqtrd 2315 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  ( A  e. 
Fin  /\  ( ran  F 
\  { 0 } )  C_  A  /\  A  C_  ( RR  \  { 0 } ) ) )  /\  x  e.  ( A  \  ( ran  F  \  { 0 } ) ) )  ->  ( x  x.  ( vol `  ( `' F " { x } ) ) )  =  0 )
51 simpr1 961 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  e.  Fin )
523, 14, 50, 51fsumss 12198 . 2  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  sum_ x  e.  ( ran  F  \  { 0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
532, 52eqtrd 2315 1  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( S.1 `  F )  =  sum_ x  e.  A  ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742   sum_csu 12158   vol *covol 18822   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1addlem4  19054  itg1climres  19069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
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