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Theorem ismbl4 40722
Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 23514, 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 40715 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
2 elpwi 4308 . . . . . . . . 9 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3 ovolcl 23466 . . . . . . . . 9 (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*)
42, 3syl 17 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ ℝ*)
54adantr 466 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ∈ ℝ*)
6 inss1 3981 . . . . . . . . . . 11 (𝑥𝐴) ⊆ 𝑥
76, 2syl5ss 3763 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
8 ovolcl 23466 . . . . . . . . . 10 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
97, 8syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
102ssdifssd 3899 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
11 ovolcl 23466 . . . . . . . . . 10 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
1210, 11syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
139, 12xaddcld 12336 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
1413adantr 466 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
152ovolsplit 40717 . . . . . . . 8 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
1615adantr 466 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
17 simpr 471 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
185, 14, 16, 17xrletrid 12191 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
1918ex 397 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) → (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
20 xrleid 12188 . . . . . . . . 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 466 . . . . . . 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 2777 . . . . . . . 8 ((vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = (vol*‘𝑥))
2524adantl 467 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = (vol*‘𝑥))
2622, 25breqtrd 4813 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
2726ex 397 . . . . 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 3128 . . 3 (∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ↔ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
3029anbi2i 609 . 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 382   = wceq 1631  wcel 2145  wral 3061  cdif 3720  cin 3722  wss 3723  𝒫 cpw 4298   class class class wbr 4787  dom cdm 5250  cfv 6030  (class class class)co 6796  cr 10141  *cxr 10279  cle 10281   +𝑒 cxad 12149  vol*covol 23450  volcvol 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8508  df-inf 8509  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-q 11997  df-rp 12036  df-xadd 12152  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fl 12801  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-ovol 23452  df-vol 23453
This theorem is referenced by:  vonvolmbl  41390
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