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Theorem itg2val 25245
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 13119 . . 3 < Or ℝ*
21supex 9457 . 2 sup(𝐿, ℝ*, < ) ∈ V
3 reex 11200 . 2 ℝ ∈ V
4 ovex 7441 . 2 (0[,]+∞) ∈ V
5 breq2 5152 . . . . . . 7 (𝑓 = 𝐹 → (𝑔r𝑓𝑔r𝐹))
65anbi1d 630 . . . . . 6 (𝑓 = 𝐹 → ((𝑔r𝑓𝑥 = (∫1𝑔)) ↔ (𝑔r𝐹𝑥 = (∫1𝑔))))
76rexbidv 3178 . . . . 5 (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))))
87abbidv 2801 . . . 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 2790 . . 3 (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))} = 𝐿)
1110supeq1d 9440 . 2 (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12 df-itg2 25137 . 2 2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8874 1 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  {cab 2709  wrex 3070   class class class wbr 5148  dom cdm 5676  wf 6539  cfv 6543  (class class class)co 7408  r cofr 7668  supcsup 9434  cr 11108  0cc0 11109  +∞cpnf 11244  *cxr 11246   < clt 11247  cle 11248  [,]cicc 13326  1citg1 25131  2citg2 25132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-itg2 25137
This theorem is referenced by:  itg2cl  25249  itg2ub  25250  itg2leub  25251  itg2addnclem  36534
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