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Theorem itg10a 19065
Description: The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
Hypotheses
Ref Expression
itg10a.1  |-  ( ph  ->  F  e.  dom  S.1 )
itg10a.2  |-  ( ph  ->  A  C_  RR )
itg10a.3  |-  ( ph  ->  ( vol * `  A )  =  0 )
itg10a.4  |-  ( (
ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  0 )
Assertion
Ref Expression
itg10a  |-  ( ph  ->  ( S.1 `  F
)  =  0 )
Distinct variable groups:    x, A    x, F    ph, x

Proof of Theorem itg10a
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 itg10a.1 . . 3  |-  ( ph  ->  F  e.  dom  S.1 )
2 itg1val 19038 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
4 i1ff 19031 . . . . . . . . . . . . . . . 16  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
51, 4syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : RR --> RR )
6 ffn 5389 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  F  Fn  RR )
75, 6syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  RR )
87adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  F  Fn  RR )
9 fniniseg 5646 . . . . . . . . . . . . 13  |-  ( F  Fn  RR  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
108, 9syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  <-> 
( x  e.  RR  /\  ( F `  x
)  =  k ) ) )
11 eldifsni 3750 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
1211ad2antlr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  =/=  0
)
13 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
14 eldif 3162 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
15 simplrr 737 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  k )
16 simpll 730 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ph )
17 itg10a.4 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  0 )
1816, 17sylan 457 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  0 )
1915, 18eqtr3d 2317 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  k  = 
0 )
2019ex 423 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  k  = 
0 ) )
2114, 20syl5bir 209 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( ( x  e.  RR  /\  -.  x  e.  A )  ->  k  =  0 ) )
2213, 21mpand 656 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( -.  x  e.  A  ->  k  =  0 ) )
2322necon1ad 2513 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( k  =/=  0  ->  x  e.  A ) )
2412, 23mpd 14 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  A
)
2524ex 423 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( ( x  e.  RR  /\  ( F `  x )  =  k )  ->  x  e.  A )
)
2610, 25sylbid 206 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  ->  x  e.  A
) )
2726ssrdv 3185 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } ) 
C_  A )
28 itg10a.2 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
2928adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  A  C_  RR )
3027, 29sstrd 3189 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } ) 
C_  RR )
31 itg10a.3 . . . . . . . . . . 11  |-  ( ph  ->  ( vol * `  A )  =  0 )
3231adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol * `  A )  =  0 )
33 ovolssnul 18846 . . . . . . . . . 10  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol
* `  A )  =  0 )  -> 
( vol * `  ( `' F " { k } ) )  =  0 )
3427, 29, 32, 33syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol * `  ( `' F " { k } ) )  =  0 )
35 nulmbl 18893 . . . . . . . . 9  |-  ( ( ( `' F " { k } ) 
C_  RR  /\  ( vol * `  ( `' F " { k } ) )  =  0 )  ->  ( `' F " { k } )  e.  dom  vol )
3630, 34, 35syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } )  e.  dom  vol )
37 mblvol 18889 . . . . . . . 8  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( vol `  ( `' F " { k } ) )  =  ( vol * `  ( `' F " { k } ) ) )
3836, 37syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol * `  ( `' F " { k } ) ) )
3938, 34eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
4039oveq2d 5874 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
41 difss 3303 . . . . . . . . 9  |-  ( ran 
F  \  { 0 } )  C_  ran  F
42 frn 5395 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
435, 42syl 15 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4441, 43syl5ss 3190 . . . . . . . 8  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
4544sselda 3180 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
4645recnd 8861 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
4746mul01d 9011 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  0 )  =  0 )
4840, 47eqtrd 2315 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
4948sumeq2dv 12176 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ k  e.  ( ran  F 
\  { 0 } ) 0 )
50 i1frn 19032 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
511, 50syl 15 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
52 ssfi 7083 . . . . . 6  |-  ( ( ran  F  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  ran  F )  ->  ( ran  F  \  { 0 } )  e.  Fin )
5351, 41, 52sylancl 643 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } )  e.  Fin )
5453olcd 382 . . . 4  |-  ( ph  ->  ( ( ran  F  \  { 0 } ) 
C_  ( ZZ>= `  0
)  \/  ( ran 
F  \  { 0 } )  e.  Fin ) )
55 sumz 12195 . . . 4  |-  ( ( ( ran  F  \  { 0 } ) 
C_  ( ZZ>= `  0
)  \/  ( ran 
F  \  { 0 } )  e.  Fin )  ->  sum_ k  e.  ( ran  F  \  {
0 } ) 0  =  0 )
5654, 55syl 15 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) 0  =  0 )
5749, 56eqtrd 2315 . 2  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
583, 57eqtrd 2315 1  |-  ( ph  ->  ( S.1 `  F
)  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {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   RRcr 8736   0cc0 8737    x. cmul 8742   ZZ>=cuz 10230   sum_csu 12158   vol
*covol 18822   volcvol 18823   S.1citg1 18970
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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-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-ovol 18824  df-vol 18825  df-itg1 18976
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