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Theorem ismbl 23046
Description: The predicate "𝐴 is Lebesgue-measurable". A set is measurable if it splits every other set 𝑥 in a "nice" way, that is, if the measure of the pieces 𝑥𝐴 and 𝑥𝐴 sum up to the measure of 𝑥 (assuming that the measure of 𝑥 is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
Assertion
Ref Expression
ismbl (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ineq2 3770 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
21fveq2d 6092 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
3 difeq2 3684 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43fveq2d 6092 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
52, 4oveq12d 6545 . . . . 5 (𝑦 = 𝐴 → ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
65eqeq2d 2620 . . . 4 (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
76ralbidv 2969 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
8 df-vol 22986 . . . . . 6 vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
98dmeqi 5234 . . . . 5 dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
10 dmres 5326 . . . . 5 dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))}) = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ dom vol*)
11 ovolf 23002 . . . . . . 7 vol*:𝒫 ℝ⟶(0[,]+∞)
1211fdmi 5951 . . . . . 6 dom vol* = 𝒫 ℝ
1312ineq2i 3773 . . . . 5 ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ dom vol*) = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ 𝒫 ℝ)
149, 10, 133eqtri 2636 . . . 4 dom vol = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ 𝒫 ℝ)
15 dfrab2 3862 . . . 4 {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ 𝒫 ℝ)
1614, 15eqtr4i 2635 . . 3 dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))}
177, 16elrab2 3333 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
18 reex 9884 . . . 4 ℝ ∈ V
1918elpw2 4750 . . 3 (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20 ffn 5944 . . . . . . 7 (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21 elpreima 6230 . . . . . . 7 (vol* Fn 𝒫 ℝ → (𝑥 ∈ (vol* “ ℝ) ↔ (𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
2211, 20, 21mp2b 10 . . . . . 6 (𝑥 ∈ (vol* “ ℝ) ↔ (𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ))
2322imbi1i 338 . . . . 5 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
24 impexp 461 . . . . 5 (((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2523, 24bitri 263 . . . 4 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2625ralbii2 2961 . . 3 (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
2719, 26anbi12i 729 . 2 ((𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2817, 27bitri 263 1 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wral 2896  {crab 2900  cdif 3537  cin 3539  wss 3540  𝒫 cpw 4108  ccnv 5027  dom cdm 5028  cres 5030  cima 5031   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  cr 9792  0cc0 9793   + caddc 9796  +∞cpnf 9928  [,]cicc 12008  vol*covol 22983  volcvol 22984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-sup 8209  df-inf 8210  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-rp 11668  df-ico 12011  df-icc 12012  df-fz 12156  df-seq 12622  df-exp 12681  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-ovol 22985  df-vol 22986
This theorem is referenced by:  ismbl2  23047  mblss  23051  mblsplit  23052  cmmbl  23054  shftmbl  23058  voliunlem2  23071
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