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Theorem ismbl3 42689
 Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 24145, but here +𝑒 is used, and the precondition (vol*‘𝑥) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Assertion
Ref Expression
ismbl3 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl3
StepHypRef Expression
1 ismbl2 24145 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2 inss1 4155 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝑥
32a1i 11 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥𝐴) ⊆ 𝑥)
4 elpwi 4506 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
54adantr 484 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
6 simpr 488 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
7 ovolsscl 24104 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
83, 5, 6, 7syl3anc 1368 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
9 difssd 4060 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥𝐴) ⊆ 𝑥)
10 ovolsscl 24104 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
119, 5, 6, 10syl3anc 1368 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
128, 11rexaddd 12622 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
1312adantlr 714 . . . . . . . 8 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
14 id 22 . . . . . . . . . 10 (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
1514imp 410 . . . . . . . . 9 ((((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
1615adantll 713 . . . . . . . 8 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
1713, 16eqbrtrd 5053 . . . . . . 7 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
182, 4sstrid 3926 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
19 ovolcl 24096 . . . . . . . . . . . . 13 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2018, 19syl 17 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
214ssdifssd 4070 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
22 ovolcl 24096 . . . . . . . . . . . . 13 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2321, 22syl 17 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2420, 23xaddcld 12689 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
25 pnfge 12520 . . . . . . . . . . 11 (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ* → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ +∞)
2624, 25syl 17 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ +∞)
2726adantr 484 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ +∞)
28 ovolf 24100 . . . . . . . . . . . . 13 vol*:𝒫 ℝ⟶(0[,]+∞)
2928ffvelrni 6832 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞))
3029adantr 484 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ (0[,]+∞))
31 simpr 488 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ¬ (vol*‘𝑥) ∈ ℝ)
32 xrge0nre 12838 . . . . . . . . . . 11 (((vol*‘𝑥) ∈ (0[,]+∞) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
3330, 31, 32syl2anc 587 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
3433eqcomd 2804 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → +∞ = (vol*‘𝑥))
3527, 34breqtrd 5057 . . . . . . . 8 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3635adantlr 714 . . . . . . 7 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3717, 36pm2.61dan 812 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3837ex 416 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
3912eqcomd 2804 . . . . . . . 8 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
40393adant2 1128 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
41 simp2 1134 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
4240, 41eqbrtrd 5053 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
43423exp 1116 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
4438, 43impbid 215 . . . 4 (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
4544ralbiia 3132 . . 3 (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
4645anbi2i 625 . 2 ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
471, 46bitri 278 1 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5031  dom cdm 5520  ‘cfv 6327  (class class class)co 7140  ℝcr 10532  0cc0 10533   + caddc 10536  +∞cpnf 10668  ℝ*cxr 10670   ≤ cle 10672   +𝑒 cxad 12500  [,]cicc 12736  vol*covol 24080  volcvol 24081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-cnex 10589  ax-resscn 10590  ax-1cn 10591  ax-icn 10592  ax-addcl 10593  ax-addrcl 10594  ax-mulcl 10595  ax-mulrcl 10596  ax-mulcom 10597  ax-addass 10598  ax-mulass 10599  ax-distr 10600  ax-i2m1 10601  ax-1ne0 10602  ax-1rid 10603  ax-rnegex 10604  ax-rrecex 10605  ax-cnre 10606  ax-pre-lttri 10607  ax-pre-lttrn 10608  ax-pre-ltadd 10609  ax-pre-mulgt0 10610  ax-pre-sup 10611 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-er 8279  df-map 8398  df-en 8500  df-dom 8501  df-sdom 8502  df-sup 8897  df-inf 8898  df-pnf 10673  df-mnf 10674  df-xr 10675  df-ltxr 10676  df-le 10677  df-sub 10868  df-neg 10869  df-div 11294  df-nn 11633  df-2 11695  df-3 11696  df-n0 11893  df-z 11977  df-uz 12239  df-q 12344  df-rp 12385  df-xadd 12503  df-ioo 12737  df-ico 12739  df-icc 12740  df-fz 12893  df-fl 13164  df-seq 13372  df-exp 13433  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-ovol 24082  df-vol 24083 This theorem is referenced by:  ismbl4  42696
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