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Theorem itg2const 21218
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 9373 . . . . . . 7  |-  RR  e.  _V
21a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  RR  e.  _V )
3 simpl3 993 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  B  e.  ( 0 [,) +oo )
)
4 1re 9385 . . . . . . . 8  |-  1  e.  RR
5 0re 9386 . . . . . . . 8  |-  0  e.  RR
64, 5keepel 3857 . . . . . . 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 4882 . . . . . . 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 2444 . . . . . 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 6336 . . . . 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 oveq2 6099 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  1  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  1 ) )
13 oveq2 6099 . . . . . . . 8  |-  ( if ( x  e.  A ,  1 ,  0 )  =  0  -> 
( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  ( B  x.  0 ) )
1412, 13ifsb 3802 . . . . . . 7  |-  ( B  x.  if ( x  e.  A ,  1 ,  0 ) )  =  if ( x  e.  A ,  ( B  x.  1 ) ,  ( B  x.  0 ) )
15 simp3 990 . . . . . . . . . . . 12  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  B  e.  ( 0 [,) +oo )
)
16 elrege0 11392 . . . . . . . . . . . 12  |-  ( B  e.  ( 0 [,) +oo )  <->  ( B  e.  RR  /\  0  <_  B ) )
1715, 16sylib 196 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( B  e.  RR  /\  0  <_  B ) )
1817simpld 459 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  B  e.  RR )
1918recnd 9412 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  B  e.  CC )
2019mulid1d 9403 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( B  x.  1 )  =  B )
2119mul01d 9568 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  ( B  x.  0 )  =  0 )
2220, 21ifeq12d 3809 . . . . . . 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 2487 . . . . . 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 4379 . . . . 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 2475 . . . 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 ) ) )
26 eqid 2443 . . . . . . 7  |-  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A , 
1 ,  0 ) )
2726i1f1 21168 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) )  e.  dom  S.1 )
28273adant3 1008 . . . . 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 21181 . . . 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 )
3025, 29eqeltrrd 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 )
3117simprd 463 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  0  <_  B
)
32 0le0 10411 . . . . . 6  |-  0  <_  0
33 breq2 4296 . . . . . . 7  |-  ( B  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  B  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
34 breq2 4296 . . . . . . 7  |-  ( 0  =  if ( x  e.  A ,  B ,  0 )  -> 
( 0  <_  0  <->  0  <_  if ( x  e.  A ,  B ,  0 ) ) )
3533, 34ifboth 3825 . . . . . 6  |-  ( ( 0  <_  B  /\  0  <_  0 )  -> 
0  <_  if (
x  e.  A ,  B ,  0 ) )
3631, 32, 35sylancl 662 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  0  <_  if ( x  e.  A ,  B ,  0 ) )
3736ralrimivw 2800 . . . 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 9339 . . . . . . 7  |-  RR  C_  CC
3938a1i 11 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  RR  C_  CC )
4018adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  B  e.  RR )
41 ifcl 3831 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  e.  RR )  ->  if ( x  e.  A ,  B , 
0 )  e.  RR )
4240, 5, 41sylancl 662 . . . . . . . 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 2799 . . . . . . 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 2443 . . . . . . . 8  |-  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) )
4544fnmpt 5537 . . . . . . 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 21151 . . . . 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 ) ) ) )
485a1i 11 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  /\  x  e.  RR )  ->  0  e.  RR )
49 fconstmpt 4882 . . . . . . 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 2444 . . . . . 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 6337 . . . . 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 ) ) )
5347, 52bitrd 253 . . . 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 ) ) )
5437, 53mpbird 232 . . 3  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR  /\  B  e.  ( 0 [,) +oo ) )  ->  0p  oR  <_  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )
55 itg2itg1 21214 . . 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 ) ) ) )
5630, 54, 55syl2anc 661 . 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 21182 . . 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 ) ) ) ) )
5825fveq2d 5695 . . 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 ) ) ) )
5926itg11 21169 . . . . 5  |-  ( ( A  e.  dom  vol  /\  ( vol `  A
)  e.  RR )  ->  ( S.1 `  (
x  e.  RR  |->  if ( x  e.  A ,  1 ,  0 ) ) )  =  ( vol `  A
) )
60593adant3 1008 . . . 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 6107 . . 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 2483 . 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 2475 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    C_ wss 3328   ifcif 3791   {csn 3877   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   dom cdm 4840    Fn wfn 5413   ` cfv 5418  (class class class)co 6091    oFcof 6318    oRcofr 6319   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    x. cmul 9287   +oocpnf 9415    <_ cle 9419   [,)cico 11302   volcvol 20947   S.1citg1 21095   S.2citg2 21096   0pc0p 21147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-disj 4263  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xadd 11090  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-xmet 17810  df-met 17811  df-ovol 20948  df-vol 20949  df-mbf 21099  df-itg1 21100  df-itg2 21101  df-0p 21148
This theorem is referenced by:  itg2const2  21219  itg2gt0  21238  itg2cnlem2  21240  iblconst  21295  itgconst  21296  itg2gt0cn  28447  bddiblnc  28462  ftc1anclem7  28473
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