Theorem ismbl 25580

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*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq2 4235	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
2	1	fveq2d 6924	. . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∩ 𝑦)) = (vol*‘(𝑥 ∩ 𝐴)))
3		difeq2 4143	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∖ 𝑦) = (𝑥 ∖ 𝐴))
4	3	fveq2d 6924	. . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∖ 𝑦)) = (vol*‘(𝑥 ∖ 𝐴)))
5	2, 4	oveq12d 7466	. . . . 5 ⊢ (𝑦 = 𝐴 → ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
6	5	eqeq2d 2751	. . . 4 ⊢ (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
7	6	ralbidv 3184	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
8		df-vol 25519	. . . . . 6 ⊢ vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))})
9	8	dmeqi 5929	. . . . 5 ⊢ dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))})
10		dmres 6041	. . . . 5 ⊢ dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ dom vol*)
11		ovolf 25536	. . . . . . 7 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
12	11	fdmi 6758	. . . . . 6 ⊢ dom vol* = 𝒫 ℝ
13	12	ineq2i 4238	. . . . 5 ⊢ ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ dom vol*) = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ)
14	9, 10, 13	3eqtri 2772	. . . 4 ⊢ dom vol = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ)
15		dfrab2 4339	. . . 4 ⊢ {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ)
16	14, 15	eqtr4i 2771	. . 3 ⊢ dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}
17	7, 16	elrab2 3711	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
18		reex 11275	. . . 4 ⊢ ℝ ∈ V
19	18	elpw2 5352	. . 3 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20		ffn 6747	. . . . . . 7 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21		elpreima 7091	. . . . . . 7 ⊢ (vol* Fn 𝒫 ℝ → (𝑥 ∈ (◡vol* “ ℝ) ↔ (𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ)))
22	11, 20, 21	mp2b 10	. . . . . 6 ⊢ (𝑥 ∈ (◡vol* “ ℝ) ↔ (𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ))
23	22	imbi1i 349	. . . . 5 ⊢ ((𝑥 ∈ (◡vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
24		impexp 450	. . . . 5 ⊢ (((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
25	23, 24	bitri 275	. . . 4 ⊢ ((𝑥 ∈ (◡vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
26	25	ralbii2 3095	. . 3 ⊢ (∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
27	19, 26	anbi12i 627	. 2 ⊢ ((𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))
28	17, 27	bitri 275	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  {crab 3443   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ^◡ccnv 5699  dom cdm 5700   ↾ cres 5702   “ cima 5703   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184   + caddc 11187  +∞cpnf 11321  [,]cicc 13410  vol*covol 25516  volcvol 25517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-icc 13414  df-fz 13568  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-ovol 25518  df-vol 25519

This theorem is referenced by:  ismbl2  25581  mblss  25585  mblsplit  25586  cmmbl  25588  shftmbl  25592  voliunlem2  25605