MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg2val Structured version   Unicode version

Theorem itg2val 22560
Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) }
Assertion
Ref Expression
itg2val  |-  ( F : RR --> ( 0 [,] +oo )  -> 
( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
)
Distinct variable group:    x, g, F
Allowed substitution hints:    L( x, g)

Proof of Theorem itg2val
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 xrltso 11429 . . 3  |-  <  Or  RR*
21supex 7974 . 2  |-  sup ( L ,  RR* ,  <  )  e.  _V
3 reex 9619 . 2  |-  RR  e.  _V
4 ovex 6324 . 2  |-  ( 0 [,] +oo )  e. 
_V
5 breq2 4421 . . . . . . 7  |-  ( f  =  F  ->  (
g  oR  <_ 
f  <->  g  oR  <_  F ) )
65anbi1d 709 . . . . . 6  |-  ( f  =  F  ->  (
( g  oR  <_  f  /\  x  =  ( S.1 `  g
) )  <->  ( g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
76rexbidv 2937 . . . . 5  |-  ( f  =  F  ->  ( E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) )  <->  E. g  e.  dom  S.1 ( g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
87abbidv 2556 . . . 4  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  { x  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) } )
9 itg2val.1 . . . 4  |-  L  =  { x  |  E. g  e.  dom  S.1 (
g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) }
108, 9syl6eqr 2479 . . 3  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  L )
1110supeq1d 7957 . 2  |-  ( f  =  F  ->  sup ( { x  |  E. g  e.  dom  S.1 (
g  oR  <_ 
f  /\  x  =  ( S.1 `  g ) ) } ,  RR* ,  <  )  =  sup ( L ,  RR* ,  <  ) )
12 df-itg2 22453 . 2  |-  S.2  =  ( f  e.  ( ( 0 [,] +oo )  ^m  RR )  |->  sup ( { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) } ,  RR* ,  <  ) )
132, 3, 4, 11, 12fvmptmap 7507 1  |-  ( F : RR --> ( 0 [,] +oo )  -> 
( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   {cab 2405   E.wrex 2774   class class class wbr 4417   dom cdm 4845   -->wf 5588   ` cfv 5592  (class class class)co 6296    oRcofr 6535   supcsup 7951   RRcr 9527   0cc0 9528   +oocpnf 9661   RR*cxr 9663    < clt 9664    <_ cle 9665   [,]cicc 11627   S.1citg1 22447   S.2citg2 22448
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-pre-lttri 9602  ax-pre-lttrn 9603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-itg2 22453
This theorem is referenced by:  itg2cl  22564  itg2ub  22565  itg2leub  22566  itg2addnclem  31697
  Copyright terms: Public domain W3C validator