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

Theorem itg2val 25763
Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
Hypothesis
Ref Expression
itg2val.1 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}
Assertion
Ref Expression
itg2val (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
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 13183 . . 3 < Or ℝ*
21supex 9503 . 2 sup(𝐿, ℝ*, < ) ∈ V
3 reex 11246 . 2 ℝ ∈ V
4 ovex 7464 . 2 (0[,]+∞) ∈ V
5 breq2 5147 . . . . . . 7 (𝑓 = 𝐹 → (𝑔r𝑓𝑔r𝐹))
65anbi1d 631 . . . . . 6 (𝑓 = 𝐹 → ((𝑔r𝑓𝑥 = (∫1𝑔)) ↔ (𝑔r𝐹𝑥 = (∫1𝑔))))
76rexbidv 3179 . . . . 5 (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))))
87abbidv 2808 . . . 4 (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))})
9 itg2val.1 . . . 4 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}
108, 9eqtr4di 2795 . . 3 (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))} = 𝐿)
1110supeq1d 9486 . 2 (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12 df-itg2 25656 . 2 2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8921 1 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  {cab 2714  wrex 3070   class class class wbr 5143  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  r cofr 7696  supcsup 9480  cr 11154  0cc0 11155  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  [,]cicc 13390  1citg1 25650  2citg2 25651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-itg2 25656
This theorem is referenced by:  itg2cl  25767  itg2ub  25768  itg2leub  25769  itg2addnclem  37678
  Copyright terms: Public domain W3C validator