Theorem ismbl4 47276

Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 25902, 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 47269	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
2		elpwi 4567	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3		ovolcl 25854	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ ℝ*)
5	4	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ∈ ℝ*)
6		inss1 4136	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
7	6, 2	sstrid 3924	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∩ 𝐴) ⊆ ℝ)
8		ovolcl 25854	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
10	2	ssdifssd 4086	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∖ 𝐴) ⊆ ℝ)
11		ovolcl 25854	. . . . . . . . . 10 ⊢ ((𝑥 ∖ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
13	9, 12	xaddcld 13471	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
14	13	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
15	2	ovolsplit 47271	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))))
16	15	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))))
17		simpr 490	. . . . . . 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 13324	. . . . . 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 418	. . . . 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 13321	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))))
21	20	adantr 486	. . . . . . 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 23	. . . . . . . . 9 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))))
23	22	eqcomd 2769	. . . . . . . 8 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) → ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘𝑥))
24	23	adantl 487	. . . . . . 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 5137	. . . . . 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 418	. . . . 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 215	. . . 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 3110	. . 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 635	. 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 278	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +e (vol*‘(𝑥 ∖ 𝐴)))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  𝒫 cpw 4560   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  (class class class)co 7451  ℝcr 11238  ℝ^*cxr 11381   ≤ cle 11383   +_e cxad 13279  vol*covol 25838  volcvol 25839

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-q 13116  df-rp 13161  df-xadd 13282  df-ioo 13520  df-ico 13522  df-icc 13523  df-fz 13680  df-fl 13972  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-ovol 25840  df-vol 25841

This theorem is used by:  vonvolmbl  47944