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Theorem itg2const 22575
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 9629 . . . . . . 7  |-  RR  e.  _V
21a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  RR  e.  _V )
3 simpl3 1010 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  B  e.  ( 0 [,) +oo )
)
4 1re 9641 . . . . . . . 8  |-  1  e.  RR
5 0re 9642 . . . . . . . 8  |-  0  e.  RR
64, 5keepel 3982 . . . . . . 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 4898 . . . . . . 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 2430 . . . . . 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 6562 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( ( RR 
X.  { B }
)  oF  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( x  e.  RR  |->  ( B  x.  if ( x  e.  A ,  1 ,  0 ) ) ) )
12 ovif2 6388 . . . . . . 7  |-  ( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  if ( x  e.  A ,  ( B  x.  1 ) ,  ( B  x.  0 ) )
13 simp3 1007 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  B  e.  ( 0 [,) +oo )
)
14 elrege0 11737 . . . . . . . . . . . 12  |-  ( B  e.  ( 0 [,) +oo )  <->  ( B  e.  RR  /\  0  <_  B ) )
1513, 14sylib 199 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( B  e.  RR  /\  0  <_  B ) )
1615simpld 460 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  B  e.  RR )
1716recnd 9668 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  B  e.  CC )
1817mulid1d 9659 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( B  x.  1 )  =  B )
1917mul01d 9831 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( B  x.  0 )  =  0 )
2018, 19ifeq12d 3935 . . . . . . 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 ) )
2112, 20syl5eq 2482 . . . . . 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 ) )
2221mpteq2dv 4513 . . . . 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 ) ) )
2311, 22eqtrd 2470 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( ( RR 
X.  { B }
)  oF  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
24 eqid 2429 . . . . . . 7  |-  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )
2524i1f1 22525 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  dom  S.1 )
26253adant3 1025 . . . . 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 )
2726, 16i1fmulc 22538 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( ( RR 
X.  { B }
)  oF  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  e. 
dom  S.1 )
2823, 27eqeltrrd 2518 . . 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 )
2915simprd 464 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  0  <_  B
)
30 0le0 10699 . . . . . 6  |-  0  <_  0
31 breq2 4430 . . . . . . 7  |-  ( B  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  B  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
32 breq2 4430 . . . . . . 7  |-  ( 0  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  0  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
3331, 32ifboth 3951 . . . . . 6  |-  ( ( 0  <_  B  /\  0  <_  0 )  -> 
0  <_  if (
x  e.  A ,  B ,  0 ) )
3429, 30, 33sylancl 666 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  0  <_  if ( x  e.  A ,  B ,  0 ) )
3534ralrimivw 2847 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  A. x  e.  RR  0  <_  if ( x  e.  A ,  B ,  0 ) )
36 ax-resscn 9595 . . . . . . 7  |-  RR  C_  CC
3736a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  RR  C_  CC )
3816adantr 466 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  B  e.  RR )
39 ifcl 3957 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  e.  RR )  ->  if ( x  e.  A ,  B , 
0 )  e.  RR )
4038, 5, 39sylancl 666 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  if ( x  e.  A ,  B ,  0 )  e.  RR )
4140ralrimiva 2846 . . . . . . 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 )
42 eqid 2429 . . . . . . . 8  |-  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )
4342fnmpt 5722 . . . . . . 7  |-  ( A. x  e.  RR  if ( x  e.  A ,  B ,  0 )  e.  RR  ->  (
x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  Fn  RR )
4441, 43syl 17 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  Fn  RR )
4537, 440pledm 22508 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( 0p  oR  <_  (
x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  <->  ( RR  X.  { 0 } )  oR  <_  (
x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) ) )
465a1i 11 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  0  e.  RR )
47 fconstmpt 4898 . . . . . . 7  |-  ( RR 
X.  { 0 } )  =  ( x  e.  RR  |->  0 )
4847a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( RR  X.  { 0 } )  =  ( x  e.  RR  |->  0 ) )
49 eqidd 2430 . . . . . 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 ) ) )
502, 46, 40, 48, 49ofrfval2 6563 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( ( RR 
X.  { 0 } )  oR  <_ 
( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  <->  A. x  e.  RR  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
5145, 50bitrd 256 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( 0p  oR  <_  (
x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  <->  A. x  e.  RR  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
5235, 51mpbird 235 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  0p  oR  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
53 itg2itg1 22571 . . 3  |-  ( ( ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  e.  dom  S.1  /\  0p  oR  <_  ( 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 ) ) ) )
5428, 52, 53syl2anc 665 . 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 ) ) ) )
5526, 16itg1mulc 22539 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( S.1 `  (
( RR  X.  { B } )  oF  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) )  =  ( B  x.  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) ) )
5623fveq2d 5885 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( S.1 `  (
( RR  X.  { B } )  oF  x.  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) ) )  =  ( S.1 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) ) )
5724itg11 22526 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  (
x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( vol `  A
) )
58573adant3 1025 . . . 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
) )
5958oveq2d 6321 . . 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 ) ) )
6055, 56, 593eqtr3d 2478 . 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 ) ) )
6154, 60eqtrd 2470 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442   ifcif 3915   {csn 4002   class class class wbr 4426    |-> cmpt 4484    X. cxp 4852   dom cdm 4854    Fn wfn 5596   ` cfv 5601  (class class class)co 6305    oFcof 6543    oRcofr 6544   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543   +oocpnf 9671    <_ cle 9675   [,)cico 11637   volcvol 22295   S.1citg1 22450   S.2citg2 22451   0pc0p 22504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-xmet 18898  df-met 18899  df-ovol 22296  df-vol 22297  df-mbf 22454  df-itg1 22455  df-itg2 22456  df-0p 22505
This theorem is referenced by:  itg2const2  22576  itg2gt0  22595  itg2cnlem2  22597  iblconst  22652  itgconst  22653  itg2gt0cn  31700  bddiblnc  31715  ftc1anclem7  31726
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