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Theorem itg2val 23714
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 12178 . . 3 < Or ℝ*
21supex 8524 . 2 sup(𝐿, ℝ*, < ) ∈ V
3 reex 10228 . 2 ℝ ∈ V
4 ovex 6822 . 2 (0[,]+∞) ∈ V
5 breq2 4788 . . . . . . 7 (𝑓 = 𝐹 → (𝑔𝑟𝑓𝑔𝑟𝐹))
65anbi1d 607 . . . . . 6 (𝑓 = 𝐹 → ((𝑔𝑟𝑓𝑥 = (∫1𝑔)) ↔ (𝑔𝑟𝐹𝑥 = (∫1𝑔))))
76rexbidv 3199 . . . . 5 (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝐹𝑥 = (∫1𝑔))))
87abbidv 2889 . . . 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 2822 . . 3 (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))} = 𝐿)
1110supeq1d 8507 . 2 (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < ))
12 df-itg2 23608 . 2 2 = (𝑓 ∈ ((0[,]+∞) ↑𝑚 ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔𝑟𝑓𝑥 = (∫1𝑔))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8045 1 (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  {cab 2756  wrex 3061   class class class wbr 4784  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6792  𝑟 cofr 7042  supcsup 8501  cr 10136  0cc0 10137  +∞cpnf 10272  *cxr 10274   < clt 10275  cle 10276  [,]cicc 12382  1citg1 23602  2citg2 23603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-pre-lttri 10211  ax-pre-lttrn 10212
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-sup 8503  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-itg2 23608
This theorem is referenced by:  itg2cl  23718  itg2ub  23719  itg2leub  23720  itg2addnclem  33786
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