Theorem ismbl 25477

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 4189	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
2	1	fveq2d 6879	. . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∩ 𝑦)) = (vol*‘(𝑥 ∩ 𝐴)))
3		difeq2 4095	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∖ 𝑦) = (𝑥 ∖ 𝐴))
4	3	fveq2d 6879	. . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∖ 𝑦)) = (vol*‘(𝑥 ∖ 𝐴)))
5	2, 4	oveq12d 7421	. . . . 5 ⊢ (𝑦 = 𝐴 → ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
6	5	eqeq2d 2746	. . . 4 ⊢ (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
7	6	ralbidv 3163	. . 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 25416	. . . . . 6 ⊢ vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))})
9	8	dmeqi 5884	. . . . 5 ⊢ dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))})
10		dmres 5999	. . . . 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 25433	. . . . . . 7 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
12	11	fdmi 6716	. . . . . 6 ⊢ dom vol* = 𝒫 ℝ
13	12	ineq2i 4192	. . . . 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 2762	. . . 4 ⊢ dom vol = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ)
15		dfrab2 4295	. . . 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 2761	. . 3 ⊢ dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}
17	7, 16	elrab2 3674	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))
18		reex 11218	. . . 4 ⊢ ℝ ∈ V
19	18	elpw2 5304	. . 3 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ)
20		ffn 6705	. . . . . . 7 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ)
21		elpreima 7047	. . . . . . 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 3078	. . 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 628	. 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 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  {crab 3415   ∖ cdif 3923   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ^◡ccnv 5653  dom cdm 5654   ↾ cres 5656   “ cima 5657   Fn wfn 6525  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403  ℝcr 11126  0cc0 11127   + caddc 11130  +∞cpnf 11264  [,]cicc 13363  vol*covol 25413  volcvol 25414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8717  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9452  df-inf 9453  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-n0 12500  df-z 12587  df-uz 12851  df-rp 13007  df-ico 13366  df-icc 13367  df-fz 13523  df-seq 14018  df-exp 14078  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-ovol 25415  df-vol 25416

This theorem is referenced by:  ismbl2  25478  mblss  25482  mblsplit  25483  cmmbl  25485  shftmbl  25489  voliunlem2  25502