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Theorem itg11 21285
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 21092 . . . . 5  |-  ( vol* `  (/) )  =  0
2 0mbl 21137 . . . . . 6  |-  (/)  e.  dom  vol
3 mblvol 21129 . . . . . 6  |-  ( (/)  e.  dom  vol  ->  ( vol `  (/) )  =  ( vol* `  (/) ) )
42, 3ax-mp 5 . . . . 5  |-  ( vol `  (/) )  =  ( vol* `  (/) )
5 itg10 21282 . . . . 5  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
61, 4, 53eqtr4ri 2491 . . . 4  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  ( vol `  (/) )
7 noel 3739 . . . . . . . . 9  |-  -.  x  e.  (/)
8 eleq2 2524 . . . . . . . . 9  |-  ( A  =  (/)  ->  ( x  e.  A  <->  x  e.  (/) ) )
97, 8mtbiri 303 . . . . . . . 8  |-  ( A  =  (/)  ->  -.  x  e.  A )
10 iffalse 3897 . . . . . . . 8  |-  ( -.  x  e.  A  ->  if ( x  e.  A ,  1 ,  0 )  =  0 )
119, 10syl 16 . . . . . . 7  |-  ( A  =  (/)  ->  if ( x  e.  A , 
1 ,  0 )  =  0 )
1211mpteq2dv 4477 . . . . . 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 4980 . . . . . 6  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
1512, 13, 143eqtr4g 2517 . . . . 5  |-  ( A  =  (/)  ->  F  =  ( RR  X.  {
0 } ) )
1615fveq2d 5793 . . . 4  |-  ( A  =  (/)  ->  ( S.1 `  F )  =  ( S.1 `  ( RR 
X.  { 0 } ) ) )
17 fveq2 5789 . . . 4  |-  ( A  =  (/)  ->  ( vol `  A )  =  ( vol `  (/) ) )
186, 16, 173eqtr4a 2518 . . 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 3744 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2113i1f1 21284 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  F  e.  dom  S.1 )
2221adantr 465 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  F  e.  dom  S.1 )
23 itg1val 21277 . . . . . . 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 21283 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  /\  ( A  e.  dom  vol 
->  ( `' F " { 1 } )  =  A ) )
2625simpli 458 . . . . . . . . . . . . 13  |-  F : RR
--> { 0 ,  1 }
27 frn 5663 . . . . . . . . . . . . 13  |-  ( F : RR --> { 0 ,  1 }  ->  ran 
F  C_  { 0 ,  1 } )
2826, 27ax-mp 5 . . . . . . . . . . . 12  |-  ran  F  C_ 
{ 0 ,  1 }
29 ssdif 3589 . . . . . . . . . . . 12  |-  ( ran 
F  C_  { 0 ,  1 }  ->  ( ran  F  \  {
0 } )  C_  ( { 0 ,  1 }  \  { 0 } ) )
3028, 29ax-mp 5 . . . . . . . . . . 11  |-  ( ran 
F  \  { 0 } )  C_  ( { 0 ,  1 }  \  { 0 } )
31 difprsnss 4107 . . . . . . . . . . 11  |-  ( { 0 ,  1 } 
\  { 0 } )  C_  { 1 }
3230, 31sstri 3463 . . . . . . . . . 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 21130 . . . . . . . . . . . . . . . 16  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
3534adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  A  C_  RR )
3635sselda 3454 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  y  e.  RR )
37 eleq1 2523 . . . . . . . . . . . . . . . 16  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
3837ifbid 3909 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  if ( x  e.  A ,  1 ,  0 )  =  if ( y  e.  A , 
1 ,  0 ) )
39 1ex 9482 . . . . . . . . . . . . . . . 16  |-  1  e.  _V
40 c0ex 9481 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
4139, 40ifex 3956 . . . . . . . . . . . . . . 15  |-  if ( y  e.  A , 
1 ,  0 )  e.  _V
4238, 13, 41fvmpt 5873 . . . . . . . . . . . . . 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 3895 . . . . . . . . . . . . . 14  |-  ( y  e.  A  ->  if ( y  e.  A ,  1 ,  0 )  =  1 )
4544adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  if (
y  e.  A , 
1 ,  0 )  =  1 )
4643, 45eqtrd 2492 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  =  1 )
47 ffn 5657 . . . . . . . . . . . . . 14  |-  ( F : RR --> { 0 ,  1 }  ->  F  Fn  RR )
4826, 47ax-mp 5 . . . . . . . . . . . . 13  |-  F  Fn  RR
49 fnfvelrn 5939 . . . . . . . . . . . . 13  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
5048, 36, 49sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( F `  y )  e.  ran  F )
5146, 50eqeltrrd 2540 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ran  F )
52 ax-1ne0 9452 . . . . . . . . . . . 12  |-  1  =/=  0
5352a1i 11 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  =/=  0 )
54 eldifsn 4098 . . . . . . . . . . 11  |-  ( 1  e.  ( ran  F  \  { 0 } )  <-> 
( 1  e.  ran  F  /\  1  =/=  0
) )
5551, 53, 54sylanbrc 664 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  1  e.  ( ran  F  \  {
0 } ) )
5655snssd 4116 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  { 1 }  C_  ( ran  F  \  { 0 } ) )
5733, 56eqssd 3471 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( ran  F 
\  { 0 } )  =  { 1 } )
5857sumeq1d 13280 . . . . . . 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 9486 . . . . . . . . 9  |-  1  e.  RR
6025simpri 462 . . . . . . . . . . . . . 14  |-  ( A  e.  dom  vol  ->  ( `' F " { 1 } )  =  A )
6160ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( `' F " { 1 } )  =  A )
6261fveq2d 5793 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  ( `' F " { 1 } ) )  =  ( vol `  A ) )
6362oveq2d 6206 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( 1  x.  ( vol `  A
) ) )
64 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  RR )
6564recnd 9513 . . . . . . . . . . . 12  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( vol `  A )  e.  CC )
6665mulid2d 9505 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  A
) )  =  ( vol `  A ) )
6763, 66eqtrd 2492 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  =  ( vol `  A
) )
6867, 65eqeltrd 2539 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) )  e.  CC )
69 id 22 . . . . . . . . . . 11  |-  ( z  =  1  ->  z  =  1 )
70 sneq 3985 . . . . . . . . . . . . 13  |-  ( z  =  1  ->  { z }  =  { 1 } )
7170imaeq2d 5267 . . . . . . . . . . . 12  |-  ( z  =  1  ->  ( `' F " { z } )  =  ( `' F " { 1 } ) )
7271fveq2d 5793 . . . . . . . . . . 11  |-  ( z  =  1  ->  ( vol `  ( `' F " { z } ) )  =  ( vol `  ( `' F " { 1 } ) ) )
7369, 72oveq12d 6208 . . . . . . . . . 10  |-  ( z  =  1  ->  (
z  x.  ( vol `  ( `' F " { z } ) ) )  =  ( 1  x.  ( vol `  ( `' F " { 1 } ) ) ) )
7473sumsn 13319 . . . . . . . . 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 663 . . . . . . . 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 2492 . . . . . . 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 2492 . . . . . 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 2492 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR )  /\  y  e.  A
)  ->  ( S.1 `  F )  =  ( vol `  A ) )
7978ex 434 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( y  e.  A  ->  ( S.1 `  F )  =  ( vol `  A ) ) )
8079exlimdv 1691 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( E. y 
y  e.  A  -> 
( S.1 `  F )  =  ( vol `  A
) ) )
8120, 80syl5bi 217 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( A  =/=  (/)  ->  ( S.1 `  F
)  =  ( vol `  A ) ) )
8219, 81pm2.61dne 2765 1  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  F
)  =  ( vol `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644    \ cdif 3423    C_ wss 3426   (/)c0 3735   ifcif 3889   {csn 3975   {cpr 3977    |-> cmpt 4448    X. cxp 4936   `'ccnv 4937   dom cdm 4938   ran crn 4939   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    x. cmul 9388   sum_csu 13265   vol*covol 21062   volcvol 21063   S.1citg1 21211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-xadd 11191  df-ioo 11405  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-xmet 17919  df-met 17920  df-ovol 21064  df-vol 21065  df-mbf 21215  df-itg1 21216
This theorem is referenced by:  itg2const  21334  itg2addnclem  28581
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