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Theorem ismbl3 39536
Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 23218, 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 23218 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2 inss1 3816 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝑥
32a1i 11 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥𝐴) ⊆ 𝑥)
4 elpwi 4145 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
54adantr 481 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
6 simpr 477 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
7 ovolsscl 23177 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
83, 5, 6, 7syl3anc 1323 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
9 difssd 3721 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥𝐴) ⊆ 𝑥)
10 ovolsscl 23177 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
119, 5, 6, 10syl3anc 1323 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
128, 11rexaddd 12016 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
1312adantlr 750 . . . . . . . 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 445 . . . . . . . . 9 ((((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
1615adantll 749 . . . . . . . 8 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
1713, 16eqbrtrd 4640 . . . . . . 7 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
182, 4syl5ss 3598 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
19 ovolcl 23169 . . . . . . . . . . . . 13 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2018, 19syl 17 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
214ssdifssd 3731 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
22 ovolcl 23169 . . . . . . . . . . . . 13 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2321, 22syl 17 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2420, 23xaddcld 12082 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
25 pnfge 11916 . . . . . . . . . . 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 481 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ +∞)
28 ovolf 23173 . . . . . . . . . . . . 13 vol*:𝒫 ℝ⟶(0[,]+∞)
2928ffvelrni 6319 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞))
3029adantr 481 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ (0[,]+∞))
31 simpr 477 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ¬ (vol*‘𝑥) ∈ ℝ)
32 xrge0nre 12227 . . . . . . . . . . 11 (((vol*‘𝑥) ∈ (0[,]+∞) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
3330, 31, 32syl2anc 692 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
3433eqcomd 2627 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → +∞ = (vol*‘𝑥))
3527, 34breqtrd 4644 . . . . . . . 8 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3635adantlr 750 . . . . . . 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 831 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3837ex 450 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
3912eqcomd 2627 . . . . . . . 8 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
40393adant2 1078 . . . . . . 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 1060 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
4240, 41eqbrtrd 4640 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
43423exp 1261 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
4438, 43impbid 202 . . . 4 (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
4544ralbiia 2974 . . 3 (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
4645anbi2i 729 . 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 264 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cdif 3556  cin 3558  wss 3559  𝒫 cpw 4135   class class class wbr 4618  dom cdm 5079  cfv 5852  (class class class)co 6610  cr 9887  0cc0 9888   + caddc 9891  +∞cpnf 10023  *cxr 10025  cle 10027   +𝑒 cxad 11896  [,]cicc 12128  vol*covol 23154  volcvol 23155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-xadd 11899  df-ioo 12129  df-ico 12131  df-icc 12132  df-fz 12277  df-fl 12541  df-seq 12750  df-exp 12809  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-ovol 23156  df-vol 23157
This theorem is referenced by:  ismbl4  39543
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