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Theorem itg2const 19111
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 8844 . . . . . . 7  |-  RR  e.  _V
21a1i 10 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  RR  e.  _V )
3 simpl3 960 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  B  e.  ( 0 [,)  +oo ) )
4 1re 8853 . . . . . . . 8  |-  1  e.  RR
5 0re 8854 . . . . . . . 8  |-  0  e.  RR
64, 5keepel 3635 . . . . . . 7  |-  if ( x  e.  A , 
1 ,  0 )  e.  RR
76a1i 10 . . . . . 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 4748 . . . . . . 7  |-  ( RR 
X.  { B }
)  =  ( x  e.  RR  |->  B )
98a1i 10 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( RR  X.  { B } )  =  ( x  e.  RR  |->  B ) )
10 eqidd 2297 . . . . . 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 6111 . . . . 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 5882 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  1  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  1 ) )
13 oveq2 5882 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  0  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  0 ) )
1412, 13ifsb 3587 . . . . . . 7  |-  ( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  if ( x  e.  A ,  ( B  x.  1 ) ,  ( B  x.  0 ) )
15 simp3 957 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  ( 0 [,)  +oo ) )
16 elrege0 10762 . . . . . . . . . . . 12  |-  ( B  e.  ( 0 [,) 
+oo )  <->  ( B  e.  RR  /\  0  <_  B ) )
1715, 16sylib 188 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  e.  RR  /\  0  <_  B )
)
1817simpld 445 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  RR )
1918recnd 8877 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  B  e.  CC )
2019mulid1d 8868 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  1 )  =  B )
2119mul01d 9027 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( B  x.  0 )  =  0 )
2220, 21ifeq12d 3594 . . . . . . 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 2340 . . . . . 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 4123 . . . . 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 2328 . . . 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 2296 . . . . . . 7  |-  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )
2726i1f1 19061 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  dom  S.1 )
28273adant3 975 . . . . 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 19074 . . . 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 2371 . . 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 449 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
0  <_  B )
32 0le0 9843 . . . . . 6  |-  0  <_  0
33 breq2 4043 . . . . . . 7  |-  ( B  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  B  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
34 breq2 4043 . . . . . . 7  |-  ( 0  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  0  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
3533, 34ifboth 3609 . . . . . 6  |-  ( ( 0  <_  B  /\  0  <_  0 )  -> 
0  <_  if (
x  e.  A ,  B ,  0 ) )
3631, 32, 35sylancl 643 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
0  <_  if (
x  e.  A ,  B ,  0 ) )
3736ralrimivw 2640 . . . 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 8810 . . . . . . 7  |-  RR  C_  CC
3938a1i 10 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  RR  C_  CC )
4018adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  B  e.  RR )
41 ifcl 3614 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  e.  RR )  ->  if ( x  e.  A ,  B , 
0 )  e.  RR )
4240, 5, 41sylancl 643 . . . . . . . 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 2639 . . . . . . 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 2296 . . . . . . . 8  |-  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )
4544fnmpt 5386 . . . . . . 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 15 . . . . . 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 19044 . . . . 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 10 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  /\  x  e.  RR )  ->  0  e.  RR )
49 fconstmpt 4748 . . . . . . 7  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
5049a1i 10 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  -> 
( RR  X.  {
0 } )  =  ( x  e.  RR  |->  0 ) )
51 eqidd 2297 . . . . . 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 6112 . . . . 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 244 . . . 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 223 . . 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 19107 . . 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 642 . 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 19075 . . 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 5545 . . 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 19062 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  (
x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( vol `  A
) )
60593adant3 975 . . . 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 5890 . . 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 2336 . 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 2328 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   dom cdm 4705    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092    o Rcofr 6093   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    +oocpnf 8880    <_ cle 8884   [,)cico 10674   volcvol 18839   S.1citg1 18986   S.2citg2 18987   0 pc0p 19040
This theorem is referenced by:  itg2const2  19112  itg2gt0  19131  itg2cnlem2  19133  iblconst  19188  itgconst  19189  itg2gt0cn  25006  bddiblnc  25021
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  ax-addf 8832
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-disj 4010  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-ofr 6095  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  df-itg2 18993  df-0p 19041
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