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Theorem itg11 19062
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 18868 . . . . 5  |-  ( vol
* `  (/) )  =  0
2 0mbl 18913 . . . . . 6  |-  (/)  e.  dom  vol
3 mblvol 18905 . . . . . 6  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
42, 3ax-mp 8 . . . . 5  |-  ( vol `  (/) )  =  ( vol * `  (/) )
5 itg10 19059 . . . . 5  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
61, 4, 53eqtr4ri 2327 . . . 4  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  ( vol `  (/) )
7 noel 3472 . . . . . . . . 9  |-  -.  x  e.  (/)
8 eleq2 2357 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( x  e.  A  <->  x  e.  (/) ) )
97, 8mtbiri 294 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  x  e.  A )
10 iffalse 3585 . . . . . . . 8  |-  ( -.  x  e.  A  ->  if ( x  e.  A ,  1 ,  0 )  =  0 )
119, 10syl 15 . . . . . . 7  |-  ( A  =  (/)  ->  if ( x  e.  A , 
1 ,  0 )  =  0 )
1211mpteq2dv 4123 . . . . . 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 4748 . . . . . 6  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
1512, 13, 143eqtr4g 2353 . . . . 5  |-  ( A  =  (/)  ->  F  =  ( RR  X.  {
0 } ) )
1615fveq2d 5545 . . . 4  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( S.1 `  ( RR 
X.  { 0 } ) ) )
17 fveq2 5541 . . . 4  |-  ( A  =  (/)  ->  ( vol `  A )  =  ( vol `  (/) ) )
186, 16, 173eqtr4a 2354 . . 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 3477 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2113i1f1 19061 . . . . . . . 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 19054 . . . . . . 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 19060 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  /\  ( A  e.  dom  vol 
->  ( `' F " { 1 } )  =  A ) )
2625simpli 444 . . . . . . . . . . . . 13  |-  F : RR
--> { 0 ,  1 }
27 frn 5411 . . . . . . . . . . . . 13  |-  ( F : RR --> { 0 ,  1 }  ->  ran 
F  C_  { 0 ,  1 } )
2826, 27ax-mp 8 . . . . . . . . . . . 12  |-  ran  F  C_ 
{ 0 ,  1 }
29 ssdif 3324 . . . . . . . . . . . 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 difprsnss 3769 . . . . . . . . . . 11  |-  ( { 0 ,  1 } 
\  { 0 } )  C_  { 1 }
3230, 31sstri 3201 . . . . . . . . . 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 18906 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3534adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  A  C_  RR )
3635sselda 3193 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  y  e.  RR )
37 eleq1 2356 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
3837ifbid 3596 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  if ( x  e.  A ,  1 ,  0 )  =  if ( y  e.  A , 
1 ,  0 ) )
39 1ex 8849 . . . . . . . . . . . . . . . 16  |-  1  e.  _V
40 c0ex 8848 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
4139, 40ifex 3636 . . . . . . . . . . . . . . 15  |-  if ( y  e.  A , 
1 ,  0 )  e.  _V
4238, 13, 41fvmpt 5618 . . . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  1 )
47 ffn 5405 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  ->  F  Fn  RR )
4826, 47ax-mp 8 . . . . . . . . . . . . 13  |-  F  Fn  RR
49 fnfvelrn 5678 . . . . . . . . . . . . 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 2371 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ran  F )
52 ax-1ne0 8822 . . . . . . . . . . . 12  |-  1  =/=  0
5352a1i 10 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  =/=  0 )
54 eldifsn 3762 . . . . . . . . . . 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 3776 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  { 1 }  C_  ( ran  F  \  { 0 } ) )
5733, 56eqssd 3209 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  =  { 1 } )
5857sumeq1d 12190 . . . . . . 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 8853 . . . . . . . . 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 5545 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  ( `' F " { 1 } ) )  =  ( vol `  A ) )
6362oveq2d 5890 . . . . . . . . . . 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 8877 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  CC )
6665mulid2d 8869 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  A
) )  =  ( vol `  A ) )
6763, 66eqtrd 2328 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( vol `  A
) )
6867, 65eqeltrd 2370 . . . . . . . . 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 3664 . . . . . . . . . . . . 13  |-  ( z  =  1  ->  { z }  =  { 1 } )
7170imaeq2d 5028 . . . . . . . . . . . 12  |-  ( z  =  1  ->  ( `' F " { z } )  =  ( `' F " { 1 } ) )
7271fveq2d 5545 . . . . . . . . . . 11  |-  ( z  =  1  ->  ( vol `  ( `' F " { z } ) )  =  ( vol `  ( `' F " { 1 } ) ) )
7369, 72oveq12d 5892 . . . . . . . . . 10  |-  ( z  =  1  ->  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7473sumsn 12229 . . . . . . . . 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 2328 . . . . . . 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 2328 . . . . . 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 2328 . . . . 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 1626 . . 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 2536 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653   {cpr 3654    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758   sum_csu 12174   vol *covol 18838   volcvol 18839   S.1citg1 18986
This theorem is referenced by:  itg2const  19111  itg2addnclem  25003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992
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