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Theorem ismbl 25454
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 4206 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
21fveq2d 6901 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
3 difeq2 4114 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43fveq2d 6901 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
52, 4oveq12d 7438 . . . . 5 (𝑦 = 𝐴 → ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
65eqeq2d 2739 . . . 4 (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
76ralbidv 3174 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
8 df-vol 25393 . . . . . 6 vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
98dmeqi 5907 . . . . 5 dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
10 dmres 6007 . . . . 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 25410 . . . . . . 7 vol*:𝒫 ℝ⟶(0[,]+∞)
1211fdmi 6734 . . . . . 6 dom vol* = 𝒫 ℝ
1312ineq2i 4209 . . . . 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 2760 . . . 4 dom vol = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ 𝒫 ℝ)
15 dfrab2 4311 . . . 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 2759 . . 3 dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))}
177, 16elrab2 3685 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
18 reex 11229 . . . 4 ℝ ∈ V
1918elpw2 5347 . . 3 (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20 ffn 6722 . . . . . . 7 (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21 elpreima 7067 . . . . . . 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 349 . . . . 5 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
24 impexp 450 . . . . 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 275 . . . 4 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2625ralbii2 3086 . . 3 (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
2719, 26anbi12i 627 . 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 275 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 205  wa 395   = wceq 1534  wcel 2099  {cab 2705  wral 3058  {crab 3429  cdif 3944  cin 3946  wss 3947  𝒫 cpw 4603  ccnv 5677  dom cdm 5678  cres 5680  cima 5681   Fn wfn 6543  wf 6544  cfv 6548  (class class class)co 7420  cr 11137  0cc0 11138   + caddc 11141  +∞cpnf 11275  [,]cicc 13359  vol*covol 25390  volcvol 25391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-map 8846  df-en 8964  df-dom 8965  df-sdom 8966  df-sup 9465  df-inf 9466  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-ico 13362  df-icc 13363  df-fz 13517  df-seq 13999  df-exp 14059  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-ovol 25392  df-vol 25393
This theorem is referenced by:  ismbl2  25455  mblss  25459  mblsplit  25460  cmmbl  25462  shftmbl  25466  voliunlem2  25479
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