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Theorem itg10a 19594
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 19567 . . 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 19560 . . . . . . . . . . . . . . . 16  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
51, 4syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : RR --> RR )
6 ffn 5583 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  F  Fn  RR )
75, 6syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  RR )
87adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  F  Fn  RR )
9 fniniseg 5843 . . . . . . . . . . . . 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 3920 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
1211ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  =/=  0
)
13 simprl 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
14 eldif 3322 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
15 simplrr 738 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  k )
16 simpll 731 . . . . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  0 )
1915, 18eqtr3d 2469 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  k  = 
0 )
2019ex 424 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( x  e.  ( RR  \  A
)  ->  k  = 
0 ) )
2114, 20syl5bir 210 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( ( x  e.  RR  /\  -.  x  e.  A )  ->  k  =  0 ) )
2213, 21mpand 657 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  ( -.  x  e.  A  ->  k  =  0 ) )
2322necon1ad 2665 . . . . . . . . . . . . . 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 424 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( ( x  e.  RR  /\  ( F `  x )  =  k )  ->  x  e.  A )
)
2610, 25sylbid 207 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  ->  x  e.  A
) )
2726ssrdv 3346 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } ) 
C_  A )
28 itg10a.2 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  RR )
2928adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  A  C_  RR )
3027, 29sstrd 3350 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } ) 
C_  RR )
31 itg10a.3 . . . . . . . . . . 11  |-  ( ph  ->  ( vol * `  A )  =  0 )
3231adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol * `  A )  =  0 )
33 ovolssnul 19375 . . . . . . . . . 10  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol
* `  A )  =  0 )  -> 
( vol * `  ( `' F " { k } ) )  =  0 )
3427, 29, 32, 33syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol * `  ( `' F " { k } ) )  =  0 )
35 nulmbl 19422 . . . . . . . . 9  |-  ( ( ( `' F " { k } ) 
C_  RR  /\  ( vol * `  ( `' F " { k } ) )  =  0 )  ->  ( `' F " { k } )  e.  dom  vol )
3630, 34, 35syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { k } )  e.  dom  vol )
37 mblvol 19418 . . . . . . . 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 2467 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
4039oveq2d 6089 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
41 frn 5589 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
425, 41syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4342ssdifssd 3477 . . . . . . . 8  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
4443sselda 3340 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
4544recnd 9106 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
4645mul01d 9257 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  0 )  =  0 )
4740, 46eqtrd 2467 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
4847sumeq2dv 12489 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ k  e.  ( ran  F 
\  { 0 } ) 0 )
49 i1frn 19561 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
501, 49syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
51 difss 3466 . . . . . 6  |-  ( ran 
F  \  { 0 } )  C_  ran  F
52 ssfi 7321 . . . . . 6  |-  ( ( ran  F  e.  Fin  /\  ( ran  F  \  { 0 } ) 
C_  ran  F )  ->  ( ran  F  \  { 0 } )  e.  Fin )
5350, 51, 52sylancl 644 . . . . 5  |-  ( ph  ->  ( ran  F  \  { 0 } )  e.  Fin )
5453olcd 383 . . . 4  |-  ( ph  ->  ( ( ran  F  \  { 0 } ) 
C_  ( ZZ>= `  0
)  \/  ( ran 
F  \  { 0 } )  e.  Fin ) )
55 sumz 12508 . . . 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 2467 . 2  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
583, 57eqtrd 2467 1  |-  ( ph  ->  ( S.1 `  F
)  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312   {csn 3806   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   RRcr 8981   0cc0 8982    x. cmul 8987   ZZ>=cuz 10480   sum_csu 12471   vol
*covol 19351   volcvol 19352   S.1citg1 19499
This theorem is referenced by:  itg2addnclem  26246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ovol 19353  df-vol 19354  df-itg1 19505
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