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Theorem itg1val2 19537
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 19536 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
21adantr 452 . 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 964 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( ran  F 
\  { 0 } )  C_  A )
43sselda 3316 . . . 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 965 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
65sselda 3316 . . . . . . 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 3437 . . . . . . 7  |-  ( x  e.  ( RR  \  { 0 } )  ->  x  e.  RR )
86, 7syl 16 . . . . . 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 19533 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
109adantlr 696 . . . . . . 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 457 . . . . . 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 9080 . . . . 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 9078 . . . 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 457 . . 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 19529 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
1615ad2antrr 707 . . . . . . . . 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 5558 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
18 dffn3 5565 . . . . . . . . . 10  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
1917, 18sylib 189 . . . . . . . . 9  |-  ( F : RR --> RR  ->  F : RR --> ran  F
)
2016, 19syl 16 . . . . . . . 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 3438 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
2221adantl 453 . . . . . . . . . 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 simplr3 1001 . . . . . . . . . . . . . . 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 } ) )
2423ssdifssd 3453 . . . . . . . . . . . . . 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 } ) )
25 simpr 448 . . . . . . . . . . . . . 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 } ) ) )
2624, 25sseldd 3317 . . . . . . . . . . . . 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 } ) )
27 eldifn 3438 . . . . . . . . . . . . 13  |-  ( x  e.  ( RR  \  { 0 } )  ->  -.  x  e.  { 0 } )
2826, 27syl 16 . . . . . . . . . . . 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 } )
2928biantrud 494 . . . . . . . . . . 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 } ) ) )
30 eldif 3298 . . . . . . . . . . 11  |-  ( x  e.  ( ran  F  \  { 0 } )  <-> 
( x  e.  ran  F  /\  -.  x  e. 
{ 0 } ) )
3129, 30syl6rbbr 256 . . . . . . . . . 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
) )
3222, 31mtbid 292 . . . . . . . . 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 )
33 disjsn 3836 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  F )
3432, 33sylibr 204 . . . . . . . 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 } )  =  (/) )
35 fimacnvdisj 5588 . . . . . . . 8  |-  ( ( F : RR --> ran  F  /\  ( ran  F  i^i  { x } )  =  (/) )  ->  ( `' F " { x } )  =  (/) )
3620, 34, 35syl2anc 643 . . . . . . 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 } )  =  (/) )
3736fveq2d 5699 . . . . . 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 `  (/) ) )
38 0mbl 19395 . . . . . . . 8  |-  (/)  e.  dom  vol
39 mblvol 19387 . . . . . . . 8  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
4038, 39ax-mp 8 . . . . . . 7  |-  ( vol `  (/) )  =  ( vol * `  (/) )
41 ovol0 19350 . . . . . . 7  |-  ( vol
* `  (/) )  =  0
4240, 41eqtri 2432 . . . . . 6  |-  ( vol `  (/) )  =  0
4337, 42syl6eq 2460 . . . . 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 )
4443oveq2d 6064 . . . 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 ) )
45 eldifi 3437 . . . . . . 7  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  x  e.  A
)
4645, 8sylan2 461 . . . . . 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 )
4746recnd 9078 . . . . 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 )
4847mul01d 9229 . . . 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 )
4944, 48eqtrd 2444 . . 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 )
50 simpr1 963 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  e.  Fin )
513, 14, 49, 50fsumss 12482 . 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 } ) ) ) )
522, 51eqtrd 2444 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3285    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   Fincfn 7076   CCcc 8952   RRcr 8953   0cc0 8954    x. cmul 8959   sum_csu 12442   vol *covol 19320   volcvol 19321   S.1citg1 19468
This theorem is referenced by:  itg1addlem4  19552  itg1climres  19567
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xadd 10675  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-xmet 16658  df-met 16659  df-ovol 19322  df-vol 19323  df-mbf 19473  df-itg1 19474
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