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Theorem itg11 19575
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 19381 . . . . 5  |-  ( vol
* `  (/) )  =  0
2 0mbl 19426 . . . . . 6  |-  (/)  e.  dom  vol
3 mblvol 19418 . . . . . 6  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol * `  (/) ) )
42, 3ax-mp 8 . . . . 5  |-  ( vol `  (/) )  =  ( vol * `  (/) )
5 itg10 19572 . . . . 5  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
61, 4, 53eqtr4ri 2466 . . . 4  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  ( vol `  (/) )
7 noel 3624 . . . . . . . . 9  |-  -.  x  e.  (/)
8 eleq2 2496 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( x  e.  A  <->  x  e.  (/) ) )
97, 8mtbiri 295 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  x  e.  A )
10 iffalse 3738 . . . . . . . 8  |-  ( -.  x  e.  A  ->  if ( x  e.  A ,  1 ,  0 )  =  0 )
119, 10syl 16 . . . . . . 7  |-  ( A  =  (/)  ->  if ( x  e.  A , 
1 ,  0 )  =  0 )
1211mpteq2dv 4288 . . . . . 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 4913 . . . . . 6  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
1512, 13, 143eqtr4g 2492 . . . . 5  |-  ( A  =  (/)  ->  F  =  ( RR  X.  {
0 } ) )
1615fveq2d 5724 . . . 4  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( S.1 `  ( RR 
X.  { 0 } ) ) )
17 fveq2 5720 . . . 4  |-  ( A  =  (/)  ->  ( vol `  A )  =  ( vol `  (/) ) )
186, 16, 173eqtr4a 2493 . . 3  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( vol `  A ) )
1918a1i 11 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
20 n0 3629 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2113i1f1 19574 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  F  e.  dom  S.1 )
2221adantr 452 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  F  e.  dom  S.1 )
23 itg1val 19567 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ z  e.  ( ran  F  \  {
0 } ) ( z  x.  ( vol `  ( `' F " { z } ) ) ) )
2422, 23syl 16 . . . . . 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 19573 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  /\  ( A  e.  dom  vol 
->  ( `' F " { 1 } )  =  A ) )
2625simpli 445 . . . . . . . . . . . . 13  |-  F : RR
--> { 0 ,  1 }
27 frn 5589 . . . . . . . . . . . . 13  |-  ( F : RR --> { 0 ,  1 }  ->  ran 
F  C_  { 0 ,  1 } )
2826, 27ax-mp 8 . . . . . . . . . . . 12  |-  ran  F  C_ 
{ 0 ,  1 }
29 ssdif 3474 . . . . . . . . . . . 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 3926 . . . . . . . . . . 11  |-  ( { 0 ,  1 } 
\  { 0 } )  C_  { 1 }
3230, 31sstri 3349 . . . . . . . . . 10  |-  ( ran 
F  \  { 0 } )  C_  { 1 }
3332a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  C_  { 1 } )
34 mblss 19419 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3534adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  A  C_  RR )
3635sselda 3340 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  y  e.  RR )
37 eleq1 2495 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
3837ifbid 3749 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  if ( x  e.  A ,  1 ,  0 )  =  if ( y  e.  A , 
1 ,  0 ) )
39 1ex 9078 . . . . . . . . . . . . . . . 16  |-  1  e.  _V
40 c0ex 9077 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
4139, 40ifex 3789 . . . . . . . . . . . . . . 15  |-  if ( y  e.  A , 
1 ,  0 )  e.  _V
4238, 13, 41fvmpt 5798 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  ( F `  y )  =  if ( y  e.  A ,  1 ,  0 ) )
4336, 42syl 16 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  if ( y  e.  A ,  1 ,  0 ) )
44 iftrue 3737 . . . . . . . . . . . . . 14  |-  ( y  e.  A  ->  if ( y  e.  A ,  1 ,  0 )  =  1 )
4544adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  if (
y  e.  A , 
1 ,  0 )  =  1 )
4643, 45eqtrd 2467 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  1 )
47 ffn 5583 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  ->  F  Fn  RR )
4826, 47ax-mp 8 . . . . . . . . . . . . 13  |-  F  Fn  RR
49 fnfvelrn 5859 . . . . . . . . . . . . 13  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
5048, 36, 49sylancr 645 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  ran  F )
5146, 50eqeltrrd 2510 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ran  F )
52 ax-1ne0 9051 . . . . . . . . . . . 12  |-  1  =/=  0
5352a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  =/=  0 )
54 eldifsn 3919 . . . . . . . . . . 11  |-  ( 1  e.  ( ran  F  \  { 0 } )  <-> 
( 1  e.  ran  F  /\  1  =/=  0
) )
5551, 53, 54sylanbrc 646 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ( ran  F  \  {
0 } ) )
5655snssd 3935 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  { 1 }  C_  ( ran  F  \  { 0 } ) )
5733, 56eqssd 3357 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  =  { 1 } )
5857sumeq1d 12487 . . . . . . 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 9082 . . . . . . . . 9  |-  1  e.  RR
6025simpri 449 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  vol  ->  ( `' F " { 1 } )  =  A )
6160ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( `' F " { 1 } )  =  A )
6261fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  ( `' F " { 1 } ) )  =  ( vol `  A ) )
6362oveq2d 6089 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( 1  x.  ( vol `  A
) ) )
64 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  RR )
6564recnd 9106 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  CC )
6665mulid2d 9098 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  A
) )  =  ( vol `  A ) )
6763, 66eqtrd 2467 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( vol `  A
) )
6867, 65eqeltrd 2509 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  e.  CC )
69 id 20 . . . . . . . . . . 11  |-  ( z  =  1  ->  z  =  1 )
70 sneq 3817 . . . . . . . . . . . . 13  |-  ( z  =  1  ->  { z }  =  { 1 } )
7170imaeq2d 5195 . . . . . . . . . . . 12  |-  ( z  =  1  ->  ( `' F " { z } )  =  ( `' F " { 1 } ) )
7271fveq2d 5724 . . . . . . . . . . 11  |-  ( z  =  1  ->  ( vol `  ( `' F " { z } ) )  =  ( vol `  ( `' F " { 1 } ) ) )
7369, 72oveq12d 6091 . . . . . . . . . 10  |-  ( z  =  1  ->  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7473sumsn 12526 . . . . . . . . 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 645 . . . . . . . 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 2467 . . . . . . 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 2467 . . . . . 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 2467 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( S.1 `  F )  =  ( vol `  A ) )
7978ex 424 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( y  e.  A  ->  ( S.1 `  F )  =  ( vol `  A ) ) )
8079exlimdv 1646 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( E. y 
y  e.  A  -> 
( S.1 `  F )  =  ( vol `  A
) ) )
8120, 80syl5bi 209 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =/=  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
8219, 81pm2.61dne 2675 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 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    C_ wss 3312   (/)c0 3620   ifcif 3731   {csn 3806   {cpr 3807    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987   sum_csu 12471   vol *covol 19351   volcvol 19352   S.1citg1 19499
This theorem is referenced by:  itg2const  19624  itg2addnclem  26246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xadd 10703  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-xmet 16687  df-met 16688  df-ovol 19353  df-vol 19354  df-mbf 19504  df-itg1 19505
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