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Theorem itg2const 19625
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 9074 . . . . . . 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 9083 . . . . . . . 8  |-  1  e.  RR
5 0re 9084 . . . . . . . 8  |-  0  e.  RR
64, 5keepel 3789 . . . . . . 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 4914 . . . . . . 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 2437 . . . . . 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 6315 . . . . 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 6082 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  1  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  1 ) )
13 oveq2 6082 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  0  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  0 ) )
1412, 13ifsb 3741 . . . . . . 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 11000 . . . . . . . . . . . 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 9107 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  CC )
2019mulid1d 9098 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  1 )  =  B )
2119mul01d 9258 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  0 )  =  0 )
2220, 21ifeq12d 3748 . . . . . . 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 2480 . . . . . 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 4289 . . . . 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 2468 . . . 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 2436 . . . . . . 7  |-  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )
2726i1f1 19575 . . . . . 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 19588 . . . 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 2511 . . 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 10074 . . . . . 6  |-  0  <_  0
33 breq2 4209 . . . . . . 7  |-  ( B  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  B  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
34 breq2 4209 . . . . . . 7  |-  ( 0  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  0  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
3533, 34ifboth 3763 . . . . . 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 2783 . . . 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 9040 . . . . . . 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 3768 . . . . . . . . 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 2782 . . . . . . 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 2436 . . . . . . . 8  |-  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )
4544fnmpt 5564 . . . . . . 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 19558 . . . . 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 4914 . . . . . . 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 2437 . . . . . 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 6316 . . . . 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 19621 . . 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 19589 . . 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 5725 . . 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 19576 . . . . 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 6090 . . 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 2476 . 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 2468 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 1652    e. wcel 1725   A.wral 2698   _Vcvv 2949    C_ wss 3313   ifcif 3732   {csn 3807   class class class wbr 4205    e. cmpt 4259    X. cxp 4869   dom cdm 4871    Fn wfn 5442   ` cfv 5447  (class class class)co 6074    o Fcof 6296    o Rcofr 6297   CCcc 8981   RRcr 8982   0cc0 8983   1c1 8984    x. cmul 8988    +oocpnf 9110    <_ cle 9114   [,)cico 10911   volcvol 19353   S.1citg1 19500   S.2citg2 19501   0 pc0p 19554
This theorem is referenced by:  itg2const2  19626  itg2gt0  19645  itg2cnlem2  19647  iblconst  19702  itgconst  19703  itg2gt0cn  26251  bddiblnc  26266  ftc1anclem7  26277
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-disj 4176  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-ofr 6299  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-er 6898  df-map 7013  df-pm 7014  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-oi 7472  df-card 7819  df-cda 8041  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-q 10568  df-rp 10606  df-xadd 10704  df-ioo 10913  df-ico 10915  df-icc 10916  df-fz 11037  df-fzo 11129  df-fl 11195  df-seq 11317  df-exp 11376  df-hash 11612  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-clim 12275  df-sum 12473  df-xmet 16688  df-met 16689  df-ovol 19354  df-vol 19355  df-mbf 19505  df-itg1 19506  df-itg2 19507  df-0p 19555
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