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Theorem ismbl 24133
 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 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴 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 4136 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
21fveq2d 6653 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
3 difeq2 4047 . . . . . . 7 (𝑦 = 𝐴 → (𝑥𝑦) = (𝑥𝐴))
43fveq2d 6653 . . . . . 6 (𝑦 = 𝐴 → (vol*‘(𝑥𝑦)) = (vol*‘(𝑥𝐴)))
52, 4oveq12d 7157 . . . . 5 (𝑦 = 𝐴 → ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
65eqeq2d 2812 . . . 4 (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
76ralbidv 3165 . . 3 (𝑦 = 𝐴 → (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦))) ↔ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
8 df-vol 24072 . . . . . 6 vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
98dmeqi 5741 . . . . 5 dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))})
10 dmres 5844 . . . . 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 24089 . . . . . . 7 vol*:𝒫 ℝ⟶(0[,]+∞)
1211fdmi 6502 . . . . . 6 dom vol* = 𝒫 ℝ
1312ineq2i 4139 . . . . 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 2828 . . . 4 dom vol = ({𝑦 ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))} ∩ 𝒫 ℝ)
15 dfrab2 4234 . . . 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 2827 . . 3 dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝑦)) + (vol*‘(𝑥𝑦)))}
177, 16elrab2 3634 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
18 reex 10621 . . . 4 ℝ ∈ V
1918elpw2 5215 . . 3 (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20 ffn 6491 . . . . . . 7 (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21 elpreima 6809 . . . . . . 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 353 . . . . 5 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
24 impexp 454 . . . . 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 278 . . . 4 ((𝑥 ∈ (vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))))
2625ralbii2 3134 . . 3 (∀𝑥 ∈ (vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴)))))
2719, 26anbi12i 629 . 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 278 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  {cab 2779  ∀wral 3109  {crab 3113   ∖ cdif 3881   ∩ cin 3883   ⊆ wss 3884  𝒫 cpw 4500  ◡ccnv 5522  dom cdm 5523   ↾ cres 5525   “ cima 5526   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  ℝcr 10529  0cc0 10530   + caddc 10533  +∞cpnf 10665  [,]cicc 12733  vol*covol 24069  volcvol 24070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-ico 12736  df-icc 12737  df-fz 12890  df-seq 13369  df-exp 13430  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-ovol 24071  df-vol 24072 This theorem is referenced by:  ismbl2  24134  mblss  24138  mblsplit  24139  cmmbl  24141  shftmbl  24145  voliunlem2  24158
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