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Theorem itg10a 21852
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 21825 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
4 i1ff 21818 . . . . . . . . . . . . . . . 16  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
51, 4syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : RR --> RR )
6 ffn 5729 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  F  Fn  RR )
75, 6syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  RR )
87adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  F  Fn  RR )
9 fniniseg 6000 . . . . . . . . . . . . 13  |-  ( F  Fn  RR  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
108, 9syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  <-> 
( x  e.  RR  /\  ( F `  x
)  =  k ) ) )
11 eldifsni 4153 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
1211ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  =/=  0
)
13 simprl 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
14 eldif 3486 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
15 simplrr 760 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  k )
16 simpll 753 . . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  0 )
1915, 18eqtr3d 2510 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  k  = 
0 )
2019ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  k  = 
0 ) )
2114, 20syl5bir 218 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( ( x  e.  RR  /\  -.  x  e.  A )  ->  k  =  0 ) )
2213, 21mpand 675 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( -.  x  e.  A  ->  k  =  0 ) )
2322necon1ad 2683 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( k  =/=  0  ->  x  e.  A ) )
2412, 23mpd 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  A
)
2524ex 434 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( ( x  e.  RR  /\  ( F `  x )  =  k )  ->  x  e.  A )
)
2610, 25sylbid 215 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  ->  x  e.  A
) )
2726ssrdv 3510 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } ) 
C_  A )
28 itg10a.2 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
2928adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  A  C_  RR )
3027, 29sstrd 3514 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } ) 
C_  RR )
31 itg10a.3 . . . . . . . . . . 11  |-  ( ph  ->  ( vol* `  A )  =  0 )
3231adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol* `  A )  =  0 )
33 ovolssnul 21633 . . . . . . . . . 10  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( `' F " { k } ) )  =  0 )
3427, 29, 32, 33syl3anc 1228 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol* `  ( `' F " { k } ) )  =  0 )
35 nulmbl 21681 . . . . . . . . 9  |-  ( ( ( `' F " { k } ) 
C_  RR  /\  ( vol* `  ( `' F " { k } ) )  =  0 )  ->  ( `' F " { k } )  e.  dom  vol )
3630, 34, 35syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } )  e.  dom  vol )
37 mblvol 21676 . . . . . . . 8  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
3836, 37syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
3938, 34eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
4039oveq2d 6298 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
41 frn 5735 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
425, 41syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4342ssdifssd 3642 . . . . . . . 8  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
4443sselda 3504 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
4544recnd 9618 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
4645mul01d 9774 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  0 )  =  0 )
4740, 46eqtrd 2508 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
4847sumeq2dv 13484 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ k  e.  ( ran  F 
\  { 0 } ) 0 )
49 i1frn 21819 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
501, 49syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
51 difss 3631 . . . . . 6  |-  ( ran 
F  \  { 0 } )  C_  ran  F
52 ssfi 7737 . . . . . 6  |-  ( ( ran  F  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  ran  F )  ->  ( ran  F  \  { 0 } )  e.  Fin )
5350, 51, 52sylancl 662 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } )  e.  Fin )
5453olcd 393 . . . 4  |-  ( ph  ->  ( ( ran  F  \  { 0 } ) 
C_  ( ZZ>= `  0
)  \/  ( ran 
F  \  { 0 } )  e.  Fin ) )
55 sumz 13503 . . . 4  |-  ( ( ( ran  F  \  { 0 } ) 
C_  ( ZZ>= `  0
)  \/  ( ran 
F  \  { 0 } )  e.  Fin )  ->  sum_ k  e.  ( ran  F  \  {
0 } ) 0  =  0 )
5654, 55syl 16 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) 0  =  0 )
5748, 56eqtrd 2508 . 2  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
583, 57eqtrd 2508 1  |-  ( ph  ->  ( S.1 `  F
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   {csn 4027   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   Fincfn 7513   RRcr 9487   0cc0 9488    x. cmul 9493   ZZ>=cuz 11078   sum_csu 13467   vol*covol 21609   volcvol 21610   S.1citg1 21759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-ovol 21611  df-vol 21612  df-itg1 21764
This theorem is referenced by:  itg2addnclem  29643
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