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Theorem itg11 19046
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 18852 . . . . 5  |-  ( vol
* `  (/) )  =  0
2 0mbl 18897 . . . . . 6  |-  (/)  e.  dom  vol
3 mblvol 18889 . . . . . 6  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
42, 3ax-mp 8 . . . . 5  |-  ( vol `  (/) )  =  ( vol * `  (/) )
5 itg10 19043 . . . . 5  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
61, 4, 53eqtr4ri 2314 . . . 4  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  ( vol `  (/) )
7 noel 3459 . . . . . . . . 9  |-  -.  x  e.  (/)
8 eleq2 2344 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( x  e.  A  <->  x  e.  (/) ) )
97, 8mtbiri 294 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  x  e.  A )
10 iffalse 3572 . . . . . . . 8  |-  ( -.  x  e.  A  ->  if ( x  e.  A ,  1 ,  0 )  =  0 )
119, 10syl 15 . . . . . . 7  |-  ( A  =  (/)  ->  if ( x  e.  A , 
1 ,  0 )  =  0 )
1211mpteq2dv 4107 . . . . . 6  |-  ( A  =  (/)  ->  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  0 ) )
13 i1f1.1 . . . . . 6  |-  F  =  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )
14 fconstmpt 4732 . . . . . 6  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
1512, 13, 143eqtr4g 2340 . . . . 5  |-  ( A  =  (/)  ->  F  =  ( RR  X.  {
0 } ) )
1615fveq2d 5529 . . . 4  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( S.1 `  ( RR 
X.  { 0 } ) ) )
17 fveq2 5525 . . . 4  |-  ( A  =  (/)  ->  ( vol `  A )  =  ( vol `  (/) ) )
186, 16, 173eqtr4a 2341 . . 3  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( vol `  A ) )
1918a1i 10 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
20 n0 3464 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2113i1f1 19045 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  F  e.  dom  S.1 )
2221adantr 451 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  F  e.  dom  S.1 )
23 itg1val 19038 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ z  e.  ( ran  F  \  {
0 } ) ( z  x.  ( vol `  ( `' F " { z } ) ) ) )
2422, 23syl 15 . . . . . 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 } ) ) ) )
2513i1f1lem 19044 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  /\  ( A  e.  dom  vol 
->  ( `' F " { 1 } )  =  A ) )
2625simpli 444 . . . . . . . . . . . . 13  |-  F : RR
--> { 0 ,  1 }
27 frn 5395 . . . . . . . . . . . . 13  |-  ( F : RR --> { 0 ,  1 }  ->  ran 
F  C_  { 0 ,  1 } )
2826, 27ax-mp 8 . . . . . . . . . . . 12  |-  ran  F  C_ 
{ 0 ,  1 }
29 ssdif 3311 . . . . . . . . . . . 12  |-  ( ran 
F  C_  { 0 ,  1 }  ->  ( ran  F  \  {
0 } )  C_  ( { 0 ,  1 }  \  { 0 } ) )
3028, 29ax-mp 8 . . . . . . . . . . 11  |-  ( ran 
F  \  { 0 } )  C_  ( { 0 ,  1 }  \  { 0 } )
31 difprsn 3756 . . . . . . . . . . 11  |-  ( { 0 ,  1 } 
\  { 0 } )  C_  { 1 }
3230, 31sstri 3188 . . . . . . . . . 10  |-  ( ran 
F  \  { 0 } )  C_  { 1 }
3332a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  C_  { 1 } )
34 mblss 18890 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3534adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  A  C_  RR )
3635sselda 3180 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  y  e.  RR )
37 eleq1 2343 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
3837ifbid 3583 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  if ( x  e.  A ,  1 ,  0 )  =  if ( y  e.  A , 
1 ,  0 ) )
39 1ex 8833 . . . . . . . . . . . . . . . 16  |-  1  e.  _V
40 c0ex 8832 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
4139, 40ifex 3623 . . . . . . . . . . . . . . 15  |-  if ( y  e.  A , 
1 ,  0 )  e.  _V
4238, 13, 41fvmpt 5602 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  ( F `  y )  =  if ( y  e.  A ,  1 ,  0 ) )
4336, 42syl 15 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  if ( y  e.  A ,  1 ,  0 ) )
44 iftrue 3571 . . . . . . . . . . . . . 14  |-  ( y  e.  A  ->  if ( y  e.  A ,  1 ,  0 )  =  1 )
4544adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  if (
y  e.  A , 
1 ,  0 )  =  1 )
4643, 45eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  1 )
47 ffn 5389 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  ->  F  Fn  RR )
4826, 47ax-mp 8 . . . . . . . . . . . . 13  |-  F  Fn  RR
49 fnfvelrn 5662 . . . . . . . . . . . . 13  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
5048, 36, 49sylancr 644 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  ran  F )
5146, 50eqeltrrd 2358 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ran  F )
52 ax-1ne0 8806 . . . . . . . . . . . 12  |-  1  =/=  0
5352a1i 10 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  =/=  0 )
54 eldifsn 3749 . . . . . . . . . . 11  |-  ( 1  e.  ( ran  F  \  { 0 } )  <-> 
( 1  e.  ran  F  /\  1  =/=  0
) )
5551, 53, 54sylanbrc 645 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ( ran  F  \  {
0 } ) )
5655snssd 3760 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  { 1 }  C_  ( ran  F  \  { 0 } ) )
5733, 56eqssd 3196 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  =  { 1 } )
5857sumeq1d 12174 . . . . . . 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 } ) ) ) )
59 1re 8837 . . . . . . . . 9  |-  1  e.  RR
6025simpri 448 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  vol  ->  ( `' F " { 1 } )  =  A )
6160ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( `' F " { 1 } )  =  A )
6261fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  ( `' F " { 1 } ) )  =  ( vol `  A ) )
6362oveq2d 5874 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( 1  x.  ( vol `  A
) ) )
64 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  RR )
6564recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  CC )
6665mulid2d 8853 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  A
) )  =  ( vol `  A ) )
6763, 66eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( vol `  A
) )
6867, 65eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  e.  CC )
69 id 19 . . . . . . . . . . 11  |-  ( z  =  1  ->  z  =  1 )
70 sneq 3651 . . . . . . . . . . . . 13  |-  ( z  =  1  ->  { z }  =  { 1 } )
7170imaeq2d 5012 . . . . . . . . . . . 12  |-  ( z  =  1  ->  ( `' F " { z } )  =  ( `' F " { 1 } ) )
7271fveq2d 5529 . . . . . . . . . . 11  |-  ( z  =  1  ->  ( vol `  ( `' F " { z } ) )  =  ( vol `  ( `' F " { 1 } ) ) )
7369, 72oveq12d 5876 . . . . . . . . . 10  |-  ( z  =  1  ->  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7473sumsn 12213 . . . . . . . . 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 } ) ) ) )
7559, 68, 74sylancr 644 . . . . . . . 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 } ) ) ) )
7675, 67eqtrd 2315 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  sum_ z  e. 
{ 1 }  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( vol `  A ) )
7758, 76eqtrd 2315 . . . . . 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 ) )
7824, 77eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( S.1 `  F )  =  ( vol `  A ) )
7978ex 423 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( y  e.  A  ->  ( S.1 `  F )  =  ( vol `  A ) ) )
8079exlimdv 1664 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( E. y 
y  e.  A  -> 
( S.1 `  F )  =  ( vol `  A
) ) )
8120, 80syl5bi 208 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =/=  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
8219, 81pm2.61dne 2523 1  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  F
)  =  ( vol `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   (/)c0 3455   ifcif 3565   {csn 3640   {cpr 3641    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   sum_csu 12158   vol *covol 18822   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg2const  19095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
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