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Theorem itg10a 22411
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 22384 . . 3  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
31, 2syl 17 . 2  |-  ( ph  ->  ( S.1 `  F
)  =  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
4 i1ff 22377 . . . . . . . . . . . . . . . 16  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
51, 4syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : RR --> RR )
6 ffn 5716 . . . . . . . . . . . . . . 15  |-  ( F : RR --> RR  ->  F  Fn  RR )
75, 6syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  Fn  RR )
87adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  F  Fn  RR )
9 fniniseg 5988 . . . . . . . . . . . . 13  |-  ( F  Fn  RR  ->  (
x  e.  ( `' F " { k } )  <->  ( x  e.  RR  /\  ( F `
 x )  =  k ) ) )
108, 9syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  <-> 
( x  e.  RR  /\  ( F `  x
)  =  k ) ) )
11 eldifsni 4100 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
1211ad2antlr 727 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  k  =/=  0
)
13 simprl 758 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( ran  F  \  { 0 } ) )  /\  ( x  e.  RR  /\  ( F `  x )  =  k ) )  ->  x  e.  RR )
14 eldif 3426 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( RR  \  A )  <->  ( x  e.  RR  /\  -.  x  e.  A ) )
15 simplrr 765 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  k  e.  ( ran  F 
\  { 0 } ) )  /\  (
x  e.  RR  /\  ( F `  x )  =  k ) )  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x )  =  k )
16 simpll 754 . . . . . . . . . . . . . . . . . . . 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 2447 . . . . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . 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 217 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( x  e.  ( `' F " { k } )  ->  x  e.  A
) )
2726ssrdv 3450 . . . . . . . . . 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 3454 . . . . . . . . 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 22192 . . . . . . . . . 10  |-  ( ( ( `' F " { k } ) 
C_  A  /\  A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( `' F " { k } ) )  =  0 )
3427, 29, 32, 33syl3anc 1232 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol* `  ( `' F " { k } ) )  =  0 )
35 nulmbl 22240 . . . . . . . . 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 22235 . . . . . . . 8  |-  ( ( `' F " { k } )  e.  dom  vol 
->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
3836, 37syl 17 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol* `  ( `' F " { k } ) ) )
3938, 34eqtrd 2445 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { k } ) )  =  0 )
4039oveq2d 6296 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( k  x.  0 ) )
41 frn 5722 . . . . . . . . . 10  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
425, 41syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  RR )
4342ssdifssd 3583 . . . . . . . 8  |-  ( ph  ->  ( ran  F  \  { 0 } ) 
C_  RR )
4443sselda 3444 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  RR )
4544recnd 9654 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  k  e.  CC )
4645mul01d 9815 . . . . 5  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  0 )  =  0 )
4740, 46eqtrd 2445 . . . 4  |-  ( (
ph  /\  k  e.  ( ran  F  \  {
0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
4847sumeq2dv 13676 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ k  e.  ( ran  F 
\  { 0 } ) 0 )
49 i1frn 22378 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
501, 49syl 17 . . . . . 6  |-  ( ph  ->  ran  F  e.  Fin )
51 difss 3572 . . . . . 6  |-  ( ran 
F  \  { 0 } )  C_  ran  F
52 ssfi 7777 . . . . . 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 13695 . . . 4  |-  ( ( ( ran  F  \  { 0 } ) 
C_  ( ZZ>= `  0
)  \/  ( ran 
F  \  { 0 } )  e.  Fin )  ->  sum_ k  e.  ( ran  F  \  {
0 } ) 0  =  0 )
5654, 55syl 17 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) 0  =  0 )
5748, 56eqtrd 2445 . 2  |-  ( ph  -> 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  0 )
583, 57eqtrd 2445 1  |-  ( ph  ->  ( S.1 `  F
)  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    \/ wo 368    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600    \ cdif 3413    C_ wss 3416   {csn 3974   `'ccnv 4824   dom cdm 4825   ran crn 4826   "cima 4828    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280   Fincfn 7556   RRcr 9523   0cc0 9524    x. cmul 9529   ZZ>=cuz 11129   sum_csu 13659   vol*covol 22168   volcvol 22169   S.1citg1 22318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-q 11230  df-rp 11268  df-ioo 11588  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-sum 13660  df-ovol 22170  df-vol 22171  df-itg1 22323
This theorem is referenced by:  itg2addnclem  31452
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