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Theorem ismbl 24056
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 4182 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
21fveq2d 6668 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
3 difeq2 4092 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43fveq2d 6668 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
52, 4oveq12d 7163 . . . . 5 (𝑦 = 𝐴 → ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
65eqeq2d 2832 . . . 4 (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
76ralbidv 3197 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
8 df-vol 23995 . . . . . 6 vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
98dmeqi 5767 . . . . 5 dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
10 dmres 5869 . . . . 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 24012 . . . . . . 7 vol*:𝒫 ℝ⟶(0[,]+∞)
1211fdmi 6518 . . . . . 6 dom vol* = 𝒫 ℝ
1312ineq2i 4185 . . . . 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 2848 . . . 4 dom vol = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ 𝒫 ℝ)
15 dfrab2 4278 . . . 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 2847 . . 3 dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))}
177, 16elrab2 3682 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
18 reex 10617 . . . 4 ℝ ∈ V
1918elpw2 5240 . . 3 (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20 ffn 6508 . . . . . . 7 (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21 elpreima 6821 . . . . . . 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 351 . . . . 5 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
24 impexp 451 . . . . 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 276 . . . 4 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2625ralbii2 3163 . . 3 (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
2719, 26anbi12i 626 . 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 276 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 207  wa 396   = wceq 1528  wcel 2105  {cab 2799  wral 3138  {crab 3142  cdif 3932  cin 3934  wss 3935  𝒫 cpw 4537  ccnv 5548  dom cdm 5549  cres 5551  cima 5552   Fn wfn 6344  wf 6345  cfv 6349  (class class class)co 7145  cr 10525  0cc0 10526   + caddc 10529  +∞cpnf 10661  [,]cicc 12731  vol*covol 23992  volcvol 23993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-ico 12734  df-icc 12735  df-fz 12883  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-ovol 23994  df-vol 23995
This theorem is referenced by:  ismbl2  24057  mblss  24061  mblsplit  24062  cmmbl  24064  shftmbl  24068  voliunlem2  24081
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