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Theorem itg2val 21962
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 11348 . . 3  |-  <  Or  RR*
21supex 7924 . 2  |-  sup ( L ,  RR* ,  <  )  e.  _V
3 reex 9584 . 2  |-  RR  e.  _V
4 ovex 6310 . 2  |-  ( 0 [,] +oo )  e. 
_V
5 breq2 4451 . . . . . . 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 2973 . . . . 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 2603 . . . 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 2526 . . 3  |-  ( f  =  F  ->  { x  |  E. g  e.  dom  S.1 ( g  oR  <_  f  /\  x  =  ( S.1 `  g
) ) }  =  L )
1110supeq1d 7907 . 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 21857 . 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 7456 1  |-  ( F : RR --> ( 0 [,] +oo )  -> 
( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   {cab 2452   E.wrex 2815   class class class wbr 4447   dom cdm 4999   -->wf 5584   ` cfv 5588  (class class class)co 6285    oRcofr 6524   supcsup 7901   RRcr 9492   0cc0 9493   +oocpnf 9626   RR*cxr 9628    < clt 9629    <_ cle 9630   [,]cicc 11533   S.1citg1 21851   S.2citg2 21852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-pre-lttri 9567  ax-pre-lttrn 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-itg2 21857
This theorem is referenced by:  itg2cl  21966  itg2ub  21967  itg2leub  21968  itg2addnclem  29919
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