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Theorem ismbl 23101
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 6107 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
3 difeq2 3684 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43fveq2d 6107 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
52, 4oveq12d 6567 . . . . 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 23041 . . . . . 6 vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
98dmeqi 5247 . . . . 5 dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
10 dmres 5339 . . . . 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 23057 . . . . . . 7 vol*:𝒫 ℝ⟶(0[,]+∞)
1211fdmi 5965 . . . . . 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 9906 . . . 4 ℝ ∈ V
1918elpw2 4755 . . 3 (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20 ffn 5958 . . . . . . 7 (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21 elpreima 6245 . . . . . . 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 5037  dom cdm 5038  cres 5040  cima 5041   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  cr 9814  0cc0 9815   + caddc 9818  +∞cpnf 9950  [,]cicc 12049  vol*covol 23038  volcvol 23039
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 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-icc 12053  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-ovol 23040  df-vol 23041
This theorem is referenced by:  ismbl2  23102  mblss  23106  mblsplit  23107  cmmbl  23109  shftmbl  23113  voliunlem2  23126
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