Theorem ismbl4 43424

Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 24595, but here +_𝑒 is used, and the precondition (vol*‘𝑥) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

Ref	Expression
ismbl4	⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl4

Step	Hyp	Ref	Expression
1		ismbl3 43417	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
2		elpwi 4539	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3		ovolcl 24547	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ ℝ*)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ∈ ℝ*)
6		inss1 4159	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
7	6, 2	sstrid 3928	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∩ 𝐴) ⊆ ℝ)
8		ovolcl 24547	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
10	2	ssdifssd 4073	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∖ 𝐴) ⊆ ℝ)
11		ovolcl 24547	. . . . . . . . . 10 ⊢ ((𝑥 ∖ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
13	9, 12	xaddcld 12964	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
15	2	ovolsplit 43419	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
16	15	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
17		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
18	5, 14, 16, 17	xrletrid 12818	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
19	18	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
20	13	xrleidd 12815	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
22		id 22	. . . . . . . . 9 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
23	22	eqcomd 2744	. . . . . . . 8 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘𝑥))
24	23	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘𝑥))
25	21, 24	breqtrd 5096	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
26	25	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
27	19, 26	impbid 211	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
28	27	ralbiia 3089	. . 3 ⊢ (∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ↔ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
29	28	anbi2i 622	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
30	1, 29	bitri 274	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  ℝ^*cxr 10939   ≤ cle 10941   +_𝑒 cxad 12775  vol*covol 24531  volcvol 24532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fl 13440  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-ovol 24533  df-vol 24534

This theorem is referenced by:  vonvolmbl  44089