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Theorem itg11 22649
Description: The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
i1f1.1  |-  F  =  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )
Assertion
Ref Expression
itg11  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  F
)  =  ( vol `  A ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem itg11
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovol0 22446 . . . . 5  |-  ( vol* `  (/) )  =  0
2 0mbl 22493 . . . . . 6  |-  (/)  e.  dom  vol
3 mblvol 22484 . . . . . 6  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
42, 3ax-mp 5 . . . . 5  |-  ( vol `  (/) )  =  ( vol* `  (/) )
5 itg10 22646 . . . . 5  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
61, 4, 53eqtr4ri 2484 . . . 4  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  ( vol `  (/) )
7 noel 3735 . . . . . . . . 9  |-  -.  x  e.  (/)
8 eleq2 2518 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( x  e.  A  <->  x  e.  (/) ) )
97, 8mtbiri 305 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  x  e.  A )
109iffalsed 3892 . . . . . . 7  |-  ( A  =  (/)  ->  if ( x  e.  A , 
1 ,  0 )  =  0 )
1110mpteq2dv 4490 . . . . . 6  |-  ( A  =  (/)  ->  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  0 ) )
12 i1f1.1 . . . . . 6  |-  F  =  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )
13 fconstmpt 4878 . . . . . 6  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
1411, 12, 133eqtr4g 2510 . . . . 5  |-  ( A  =  (/)  ->  F  =  ( RR  X.  {
0 } ) )
1514fveq2d 5869 . . . 4  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( S.1 `  ( RR 
X.  { 0 } ) ) )
16 fveq2 5865 . . . 4  |-  ( A  =  (/)  ->  ( vol `  A )  =  ( vol `  (/) ) )
176, 15, 163eqtr4a 2511 . . 3  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( vol `  A ) )
1817a1i 11 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
19 n0 3741 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2012i1f1 22648 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  F  e.  dom  S.1 )
2120adantr 467 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  F  e.  dom  S.1 )
22 itg1val 22641 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ z  e.  ( ran  F  \  {
0 } ) ( z  x.  ( vol `  ( `' F " { z } ) ) ) )
2321, 22syl 17 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( S.1 `  F )  =  sum_ z  e.  ( ran  F 
\  { 0 } ) ( z  x.  ( vol `  ( `' F " { z } ) ) ) )
2412i1f1lem 22647 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  /\  ( A  e.  dom  vol 
->  ( `' F " { 1 } )  =  A ) )
2524simpli 460 . . . . . . . . . . . . 13  |-  F : RR
--> { 0 ,  1 }
26 frn 5735 . . . . . . . . . . . . 13  |-  ( F : RR --> { 0 ,  1 }  ->  ran 
F  C_  { 0 ,  1 } )
2725, 26ax-mp 5 . . . . . . . . . . . 12  |-  ran  F  C_ 
{ 0 ,  1 }
28 ssdif 3568 . . . . . . . . . . . 12  |-  ( ran 
F  C_  { 0 ,  1 }  ->  ( ran  F  \  {
0 } )  C_  ( { 0 ,  1 }  \  { 0 } ) )
2927, 28ax-mp 5 . . . . . . . . . . 11  |-  ( ran 
F  \  { 0 } )  C_  ( { 0 ,  1 }  \  { 0 } )
30 difprsnss 4107 . . . . . . . . . . 11  |-  ( { 0 ,  1 } 
\  { 0 } )  C_  { 1 }
3129, 30sstri 3441 . . . . . . . . . 10  |-  ( ran 
F  \  { 0 } )  C_  { 1 }
3231a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  C_  { 1 } )
33 mblss 22485 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3433adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  A  C_  RR )
3534sselda 3432 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  y  e.  RR )
36 eleq1 2517 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
3736ifbid 3903 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  if ( x  e.  A ,  1 ,  0 )  =  if ( y  e.  A , 
1 ,  0 ) )
38 1ex 9638 . . . . . . . . . . . . . . . 16  |-  1  e.  _V
39 c0ex 9637 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
4038, 39ifex 3949 . . . . . . . . . . . . . . 15  |-  if ( y  e.  A , 
1 ,  0 )  e.  _V
4137, 12, 40fvmpt 5948 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  ( F `  y )  =  if ( y  e.  A ,  1 ,  0 ) )
4235, 41syl 17 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  if ( y  e.  A ,  1 ,  0 ) )
43 iftrue 3887 . . . . . . . . . . . . . 14  |-  ( y  e.  A  ->  if ( y  e.  A ,  1 ,  0 )  =  1 )
4443adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  if (
y  e.  A , 
1 ,  0 )  =  1 )
4542, 44eqtrd 2485 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  1 )
46 ffn 5728 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  ->  F  Fn  RR )
4725, 46ax-mp 5 . . . . . . . . . . . . 13  |-  F  Fn  RR
48 fnfvelrn 6019 . . . . . . . . . . . . 13  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
4947, 35, 48sylancr 669 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  ran  F )
5045, 49eqeltrrd 2530 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ran  F )
51 ax-1ne0 9608 . . . . . . . . . . . 12  |-  1  =/=  0
5251a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  =/=  0 )
53 eldifsn 4097 . . . . . . . . . . 11  |-  ( 1  e.  ( ran  F  \  { 0 } )  <-> 
( 1  e.  ran  F  /\  1  =/=  0
) )
5450, 52, 53sylanbrc 670 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ( ran  F  \  {
0 } ) )
5554snssd 4117 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  { 1 }  C_  ( ran  F  \  { 0 } ) )
5632, 55eqssd 3449 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  =  { 1 } )
5756sumeq1d 13767 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  sum_ z  e.  ( ran  F  \  { 0 } ) ( z  x.  ( vol `  ( `' F " { z } ) ) )  =  sum_ z  e.  { 1 }  ( z  x.  ( vol `  ( `' F " { z } ) ) ) )
58 1re 9642 . . . . . . . . 9  |-  1  e.  RR
5924simpri 464 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  vol  ->  ( `' F " { 1 } )  =  A )
6059ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( `' F " { 1 } )  =  A )
6160fveq2d 5869 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  ( `' F " { 1 } ) )  =  ( vol `  A ) )
6261oveq2d 6306 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( 1  x.  ( vol `  A
) ) )
63 simplr 762 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  RR )
6463recnd 9669 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  CC )
6564mulid2d 9661 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  A
) )  =  ( vol `  A ) )
6662, 65eqtrd 2485 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( vol `  A
) )
6766, 64eqeltrd 2529 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  e.  CC )
68 id 22 . . . . . . . . . . 11  |-  ( z  =  1  ->  z  =  1 )
69 sneq 3978 . . . . . . . . . . . . 13  |-  ( z  =  1  ->  { z }  =  { 1 } )
7069imaeq2d 5168 . . . . . . . . . . . 12  |-  ( z  =  1  ->  ( `' F " { z } )  =  ( `' F " { 1 } ) )
7170fveq2d 5869 . . . . . . . . . . 11  |-  ( z  =  1  ->  ( vol `  ( `' F " { z } ) )  =  ( vol `  ( `' F " { 1 } ) ) )
7268, 71oveq12d 6308 . . . . . . . . . 10  |-  ( z  =  1  ->  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7372sumsn 13807 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  e.  CC )  ->  sum_ z  e.  {
1 }  ( z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7458, 67, 73sylancr 669 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  sum_ z  e. 
{ 1 }  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7574, 66eqtrd 2485 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  sum_ z  e. 
{ 1 }  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( vol `  A ) )
7657, 75eqtrd 2485 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  sum_ z  e.  ( ran  F  \  { 0 } ) ( z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( vol `  A ) )
7723, 76eqtrd 2485 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( S.1 `  F )  =  ( vol `  A ) )
7877ex 436 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( y  e.  A  ->  ( S.1 `  F )  =  ( vol `  A ) ) )
7978exlimdv 1779 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( E. y 
y  e.  A  -> 
( S.1 `  F )  =  ( vol `  A
) ) )
8019, 79syl5bi 221 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =/=  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
8118, 80pm2.61dne 2710 1  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  F
)  =  ( vol `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622    \ cdif 3401    C_ wss 3404   (/)c0 3731   ifcif 3881   {csn 3968   {cpr 3970    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   sum_csu 13752   vol*covol 22413   volcvol 22415   S.1citg1 22573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-xmet 18963  df-met 18964  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578
This theorem is referenced by:  itg2const  22698  itg2addnclem  31993
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