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Theorem itg2const 19499
Description: Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
itg2const  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( B  x.  ( vol `  A ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem itg2const
StepHypRef Expression
1 reex 9014 . . . . . . 7  |-  RR  e.  _V
21a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  RR  e.  _V )
3 simpl3 962 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  B  e.  ( 0 [,)  +oo ) )
4 1re 9023 . . . . . . . 8  |-  1  e.  RR
5 0re 9024 . . . . . . . 8  |-  0  e.  RR
64, 5keepel 3739 . . . . . . 7  |-  if ( x  e.  A , 
1 ,  0 )  e.  RR
76a1i 11 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  if ( x  e.  A ,  1 ,  0 )  e.  RR )
8 fconstmpt 4861 . . . . . . 7  |-  ( RR 
X.  { B }
)  =  ( x  e.  RR  |->  B )
98a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( RR  X.  { B } )  =  ( x  e.  RR  |->  B ) )
10 eqidd 2388 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )
112, 3, 7, 9, 10offval2 6261 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( ( RR  X.  { B } )  o F  x.  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) ) )  =  ( x  e.  RR  |->  ( B  x.  if ( x  e.  A , 
1 ,  0 ) ) ) )
12 oveq2 6028 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  1  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  1 ) )
13 oveq2 6028 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  0  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  0 ) )
1412, 13ifsb 3691 . . . . . . 7  |-  ( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  if ( x  e.  A ,  ( B  x.  1 ) ,  ( B  x.  0 ) )
15 simp3 959 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  ( 0 [,)  +oo ) )
16 elrege0 10939 . . . . . . . . . . . 12  |-  ( B  e.  ( 0 [,) 
+oo )  <->  ( B  e.  RR  /\  0  <_  B ) )
1715, 16sylib 189 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  e.  RR  /\  0  <_  B )
)
1817simpld 446 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  RR )
1918recnd 9047 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  CC )
2019mulid1d 9038 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  1 )  =  B )
2119mul01d 9197 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  0 )  =  0 )
2220, 21ifeq12d 3698 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  if ( x  e.  A ,  ( B  x.  1 ) ,  ( B  x.  0 ) )  =  if ( x  e.  A ,  B ,  0 ) )
2314, 22syl5eq 2431 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  if ( x  e.  A ,  B ,  0 ) )
2423mpteq2dv 4237 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( x  e.  RR  |->  ( B  x.  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
2511, 24eqtrd 2419 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( ( RR  X.  { B } )  o F  x.  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
26 eqid 2387 . . . . . . 7  |-  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )
2726i1f1 19449 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  dom  S.1 )
28273adant3 977 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  dom  S.1 )
2928, 18i1fmulc 19462 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( ( RR  X.  { B } )  o F  x.  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) ) )  e.  dom  S.1 )
3025, 29eqeltrrd 2462 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  e.  dom  S.1 )
3117simprd 450 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
0  <_  B )
32 0le0 10013 . . . . . 6  |-  0  <_  0
33 breq2 4157 . . . . . . 7  |-  ( B  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  B  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
34 breq2 4157 . . . . . . 7  |-  ( 0  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  0  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
3533, 34ifboth 3713 . . . . . 6  |-  ( ( 0  <_  B  /\  0  <_  0 )  -> 
0  <_  if (
x  e.  A ,  B ,  0 ) )
3631, 32, 35sylancl 644 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
0  <_  if (
x  e.  A ,  B ,  0 ) )
3736ralrimivw 2733 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  A. x  e.  RR  0  <_  if ( x  e.  A ,  B ,  0 ) )
38 ax-resscn 8980 . . . . . . 7  |-  RR  C_  CC
3938a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  RR  C_  CC )
4018adantr 452 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  B  e.  RR )
41 ifcl 3718 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  e.  RR )  ->  if ( x  e.  A ,  B , 
0 )  e.  RR )
4240, 5, 41sylancl 644 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  if ( x  e.  A ,  B , 
0 )  e.  RR )
4342ralrimiva 2732 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  A. x  e.  RR  if ( x  e.  A ,  B ,  0 )  e.  RR )
44 eqid 2387 . . . . . . . 8  |-  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )
4544fnmpt 5511 . . . . . . 7  |-  ( A. x  e.  RR  if ( x  e.  A ,  B ,  0 )  e.  RR  ->  (
x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  Fn  RR )
4643, 45syl 16 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  Fn  RR )
4739, 460pledm 19432 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( 0 p  o R  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  <-> 
( RR  X.  {
0 } )  o R  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) ) )
485a1i 11 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  0  e.  RR )
49 fconstmpt 4861 . . . . . . 7  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
5049a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( RR  X.  {
0 } )  =  ( x  e.  RR  |->  0 ) )
51 eqidd 2388 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
522, 48, 42, 50, 51ofrfval2 6262 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( ( RR  X.  { 0 } )  o R  <_  (
x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  <->  A. x  e.  RR  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
5347, 52bitrd 245 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( 0 p  o R  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  <->  A. x  e.  RR  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
5437, 53mpbird 224 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
0 p  o R  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
55 itg2itg1 19495 . . 3  |-  ( ( ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  e.  dom  S.1  /\  0 p  o R  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  ->  ( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) ) )
5630, 54, 55syl2anc 643 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) ) )
5728, 18itg1mulc 19463 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.1 `  ( ( RR  X.  { B } )  o F  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) )  =  ( B  x.  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) ) )
5825fveq2d 5672 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.1 `  ( ( RR  X.  { B } )  o F  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) )  =  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) ) )
5926itg11 19450 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  (
x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( vol `  A
) )
60593adant3 977 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.1 `  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) ) )  =  ( vol `  A ) )
6160oveq2d 6036 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) )  =  ( B  x.  ( vol `  A ) ) )
6257, 58, 613eqtr3d 2427 . 2  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( B  x.  ( vol `  A ) ) )
6356, 62eqtrd 2419 1  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( B  x.  ( vol `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    C_ wss 3263   ifcif 3682   {csn 3757   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   dom cdm 4818    Fn wfn 5389   ` cfv 5394  (class class class)co 6020    o Fcof 6242    o Rcofr 6243   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    +oocpnf 9050    <_ cle 9054   [,)cico 10850   volcvol 19227   S.1citg1 19374   S.2citg2 19375   0 pc0p 19428
This theorem is referenced by:  itg2const2  19500  itg2gt0  19519  itg2cnlem2  19521  iblconst  19576  itgconst  19577  itg2gt0cn  25960  bddiblnc  25975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-xadd 10643  df-ioo 10852  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-xmet 16619  df-met 16620  df-ovol 19228  df-vol 19229  df-mbf 19379  df-itg1 19380  df-itg2 19381  df-0p 19429
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