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Theorem itg2val 23401
 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 11918 . . 3 < Or ℝ*
21supex 8313 . 2 sup(𝐿, ℝ*, < ) ∈ V
3 reex 9971 . 2 ℝ ∈ V
4 ovex 6632 . 2 (0[,]+∞) ∈ V
5 breq2 4617 . . . . . . 7 (𝑓 = 𝐹 → (𝑔𝑟𝑓𝑔𝑟𝐹))
65anbi1d 740 . . . . . 6 (𝑓 = 𝐹 → ((𝑔𝑟𝑓𝑥 = (∫1𝑔)) ↔ (𝑔𝑟𝐹𝑥 = (∫1𝑔))))
76rexbidv 3045 . . . . 5 (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))))
87abbidv 2738 . . . 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, 9syl6eqr 2673 . . 3 (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))} = 𝐿)
1110supeq1d 8296 . 2 (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12 df-itg2 23296 . 2 2 = (𝑓 ∈ ((0[,]+∞) ↑𝑚 ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 7838 1 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480  {cab 2607  ∃wrex 2908   class class class wbr 4613  dom cdm 5074  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ∘𝑟 cofr 6849  supcsup 8290  ℝcr 9879  0cc0 9880  +∞cpnf 10015  ℝ*cxr 10017   < clt 10018   ≤ cle 10019  [,]cicc 12120  ∫1citg1 23290  ∫2citg2 23291 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-itg2 23296 This theorem is referenced by:  itg2cl  23405  itg2ub  23406  itg2leub  23407  itg2addnclem  33090
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