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Theorem itg2val 21218
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 11130 . . 3  |-  <  Or  RR*
21supex 7725 . 2  |-  sup ( L ,  RR* ,  <  )  e.  _V
3 reex 9385 . 2  |-  RR  e.  _V
4 ovex 6128 . 2  |-  ( 0 [,] +oo )  e. 
_V
5 breq2 4308 . . . . . . 7  |-  ( f  =  F  ->  (
g  oR  <_ 
f  <->  g  oR  <_  F ) )
65anbi1d 704 . . . . . 6  |-  ( f  =  F  ->  (
( g  oR  <_  f  /\  x  =  ( S.1 `  g
) )  <->  ( g  oR  <_  F  /\  x  =  ( S.1 `  g ) ) ) )
76rexbidv 2748 . . . . 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 2563 . . . 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 2493 . . 3  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  L )
1110supeq1d 7708 . 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 21113 . 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 7261 1  |-  ( F : RR --> ( 0 [,] +oo )  -> 
( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   {cab 2429   E.wrex 2728   class class class wbr 4304   dom cdm 4852   -->wf 5426   ` cfv 5430  (class class class)co 6103    oRcofr 6331   supcsup 7702   RRcr 9293   0cc0 9294   +oocpnf 9427   RR*cxr 9429    < clt 9430    <_ cle 9431   [,]cicc 11315   S.1citg1 21107   S.2citg2 21108
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-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-pre-lttri 9368  ax-pre-lttrn 9369
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-itg2 21113
This theorem is referenced by:  itg2cl  21222  itg2ub  21223  itg2leub  21224  itg2addnclem  28455
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