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Theorem ismbl4 40732
Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 23515, 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*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl4
StepHypRef Expression
1 ismbl3 40725 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
2 elpwi 4313 . . . . . . . . 9 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3 ovolcl 23467 . . . . . . . . 9 (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*)
42, 3syl 17 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ ℝ*)
54adantr 472 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ∈ ℝ*)
6 inss1 3977 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
76, 2syl5ss 3756 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
8 ovolcl 23467 . . . . . . . . . 10 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
97, 8syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
102ssdifssd 3892 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
11 ovolcl 23467 . . . . . . . . . 10 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
1210, 11syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
139, 12xaddcld 12345 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
1413adantr 472 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
152ovolsplit 40727 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
1615adantr 472 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
17 simpr 479 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
185, 14, 16, 17xrletrid 12200 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
1918ex 449 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
20 xrleid 12197 . . . . . . . . 9 (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ* → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
2113, 20syl 17 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
2221adantr 472 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
23 id 22 . . . . . . . . 9 ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
2423eqcomd 2767 . . . . . . . 8 ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = (vol*‘𝑥))
2524adantl 473 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = (vol*‘𝑥))
2622, 25breqtrd 4831 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
2726ex 449 . . . . 5 (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
2819, 27impbid 202 . . . 4 (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ↔ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
2928ralbiia 3118 . . 3 (∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ↔ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
3029anbi2i 732 . 2 ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
311, 30bitri 264 1 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2140  wral 3051  cdif 3713  cin 3715  wss 3716  𝒫 cpw 4303   class class class wbr 4805  dom cdm 5267  cfv 6050  (class class class)co 6815  cr 10148  *cxr 10286  cle 10288   +𝑒 cxad 12158  vol*covol 23452  volcvol 23453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-pre-sup 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-sup 8516  df-inf 8517  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-nn 11234  df-2 11292  df-3 11293  df-n0 11506  df-z 11591  df-uz 11901  df-q 12003  df-rp 12047  df-xadd 12161  df-ioo 12393  df-ico 12395  df-icc 12396  df-fz 12541  df-fl 12808  df-seq 13017  df-exp 13076  df-cj 14059  df-re 14060  df-im 14061  df-sqrt 14195  df-abs 14196  df-ovol 23454  df-vol 23455
This theorem is referenced by:  vonvolmbl  41400
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