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Theorem itg1val2 21174
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 21173 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ x  e.  ( ran  F  \  {
0 } ) ( x  x.  ( vol `  ( `' F " { x } ) ) ) )
21adantr 465 . 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 995 . . 3  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  ( ran  F 
\  { 0 } )  C_  A )
43sselda 3368 . . . 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 996 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  ( A  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  A  /\  A  C_  ( RR  \  {
0 } ) ) )  ->  A  C_  ( RR  \  { 0 } ) )
65sselda 3368 . . . . . . 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 3490 . . . . . . 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 21170 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  x  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
109adantlr 714 . . . . . . 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 470 . . . . . 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 9426 . . . . 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 9424 . . . 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 470 . . 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 21166 . . . . . . . . . 10  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
1615ad2antrr 725 . . . . . . . . 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 5571 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  F  Fn  RR )
18 dffn3 5578 . . . . . . . . . 10  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
1917, 18sylib 196 . . . . . . . . 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 3491 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  -.  x  e.  ( ran  F  \  {
0 } ) )
2221adantl 466 . . . . . . . . . 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 1032 . . . . . . . . . . . . . . 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 3506 . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . 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 3369 . . . . . . . . . . . . 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 3491 . . . . . . . . . . . . 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 507 . . . . . . . . . . 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 3350 . . . . . . . . . . 11  |-  ( x  e.  ( ran  F  \  { 0 } )  <-> 
( x  e.  ran  F  /\  -.  x  e. 
{ 0 } ) )
3129, 30syl6rbbr 264 . . . . . . . . . 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 300 . . . . . . . . 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 3948 . . . . . . . . 9  |-  ( ( ran  F  i^i  {
x } )  =  (/) 
<->  -.  x  e.  ran  F )
3432, 33sylibr 212 . . . . . . . 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 5601 . . . . . . . 8  |-  ( ( F : RR --> ran  F  /\  ( ran  F  i^i  { x } )  =  (/) )  ->  ( `' F " { x } )  =  (/) )
3620, 34, 35syl2anc 661 . . . . . . 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 5707 . . . . . 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 21033 . . . . . . . 8  |-  (/)  e.  dom  vol
39 mblvol 21025 . . . . . . . 8  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
4038, 39ax-mp 5 . . . . . . 7  |-  ( vol `  (/) )  =  ( vol* `  (/) )
41 ovol0 20988 . . . . . . 7  |-  ( vol* `  (/) )  =  0
4240, 41eqtri 2463 . . . . . 6  |-  ( vol `  (/) )  =  0
4337, 42syl6eq 2491 . . . . 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 6119 . . . 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 3490 . . . . . . 7  |-  ( x  e.  ( A  \ 
( ran  F  \  {
0 } ) )  ->  x  e.  A
)
4645, 8sylan2 474 . . . . . 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 9424 . . . . 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 9580 . . . 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 2475 . . 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 994 . . 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 13214 . 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 2475 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3337    i^i cin 3339    C_ wss 3340   (/)c0 3649   {csn 3889   `'ccnv 4851   dom cdm 4852   ran crn 4853   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   Fincfn 7322   CCcc 9292   RRcr 9293   0cc0 9294    x. cmul 9299   sum_csu 13175   vol*covol 20958   volcvol 20959   S.1citg1 21107
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-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-xadd 11102  df-ioo 11316  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-sum 13176  df-xmet 17822  df-met 17823  df-ovol 20960  df-vol 20961  df-mbf 21111  df-itg1 21112
This theorem is referenced by:  itg1addlem4  21189  itg1climres  21204
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