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Theorem itg10a 19469
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 19442 . . 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 19435 . . . . . . . . . . . . . . . 16  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
51, 4syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : RR --> RR )
6 ffn 5531 . . . . . . . . . . . . . . 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 5790 . . . . . . . . . . . . 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 3871 . . . . . . . . . . . . . . 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 3273 . . . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . . . . 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 2617 . . . . . . . . . . . . . 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 3297 . . . . . . . . . 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 3301 . . . . . . . . 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 19250 . . . . . . . . . 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 19297 . . . . . . . . 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 19293 . . . . . . . 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 2419 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
4039oveq2d 6036 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
41 frn 5537 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
425, 41syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4342ssdifssd 3428 . . . . . . . 8  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
4443sselda 3291 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
4544recnd 9047 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
4645mul01d 9197 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  0 )  =  0 )
4740, 46eqtrd 2419 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
4847sumeq2dv 12424 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ k  e.  ( ran  F 
\  { 0 } ) 0 )
49 i1frn 19436 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
501, 49syl 16 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
51 difss 3417 . . . . . 6  |-  ( ran 
F  \  { 0 } )  C_  ran  F
52 ssfi 7265 . . . . . 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 12443 . . . 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 2419 . 2  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
583, 57eqtrd 2419 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 1649    e. wcel 1717    =/= wne 2550    \ cdif 3260    C_ wss 3263   {csn 3757   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   Fincfn 7045   RRcr 8922   0cc0 8923    x. cmul 8928   ZZ>=cuz 10420   sum_csu 12406   vol
*covol 19226   volcvol 19227   S.1citg1 19374
This theorem is referenced by:  itg2addnclem  25957
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-ioo 10852  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-ovol 19228  df-vol 19229  df-itg1 19380
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