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Theorem itg2val 25777
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 13179 . . 3 < Or ℝ*
21supex 9500 . 2 sup(𝐿, ℝ*, < ) ∈ V
3 reex 11243 . 2 ℝ ∈ V
4 ovex 7463 . 2 (0[,]+∞) ∈ V
5 breq2 5151 . . . . . . 7 (𝑓 = 𝐹 → (𝑔r𝑓𝑔r𝐹))
65anbi1d 631 . . . . . 6 (𝑓 = 𝐹 → ((𝑔r𝑓𝑥 = (∫1𝑔)) ↔ (𝑔r𝐹𝑥 = (∫1𝑔))))
76rexbidv 3176 . . . . 5 (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))))
87abbidv 2805 . . . 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 2792 . . 3 (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))} = 𝐿)
1110supeq1d 9483 . 2 (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12 df-itg2 25669 . 2 2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8919 1 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  {cab 2711  wrex 3067   class class class wbr 5147  dom cdm 5688  wf 6558  cfv 6562  (class class class)co 7430  r cofr 7695  supcsup 9477  cr 11151  0cc0 11152  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  [,]cicc 13386  1citg1 25663  2citg2 25664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-itg2 25669
This theorem is referenced by:  itg2cl  25781  itg2ub  25782  itg2leub  25783  itg2addnclem  37657
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