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Theorem ismbl3 46001
Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 25562, 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 25562 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
2 inss1 4237 . . . . . . . . . . . 12 (𝑥𝐴) ⊆ 𝑥
32a1i 11 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥𝐴) ⊆ 𝑥)
4 elpwi 4607 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
54adantr 480 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
6 simpr 484 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
7 ovolsscl 25521 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
83, 5, 6, 7syl3anc 1373 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
9 difssd 4137 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥𝐴) ⊆ 𝑥)
10 ovolsscl 25521 . . . . . . . . . . 11 (((𝑥𝐴) ⊆ 𝑥𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
119, 5, 6, 10syl3anc 1373 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥𝐴)) ∈ ℝ)
128, 11rexaddd 13276 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))))
1312adantlr 715 . . . . . . . 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 406 . . . . . . . . 9 ((((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
1615adantll 714 . . . . . . . 8 (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
1713, 16eqbrtrd 5165 . . . . . . 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 3995 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
19 ovolcl 25513 . . . . . . . . . . . . 13 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2018, 19syl 17 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
214ssdifssd 4147 . . . . . . . . . . . . 13 (𝑥 ∈ 𝒫 ℝ → (𝑥𝐴) ⊆ ℝ)
22 ovolcl 25513 . . . . . . . . . . . . 13 ((𝑥𝐴) ⊆ ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2321, 22syl 17 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥𝐴)) ∈ ℝ*)
2420, 23xaddcld 13343 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ∈ ℝ*)
25 pnfge 13172 . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ +∞)
28 ovolf 25517 . . . . . . . . . . . . 13 vol*:𝒫 ℝ⟶(0[,]+∞)
2928ffvelcdmi 7103 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞))
3029adantr 480 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ (0[,]+∞))
31 simpr 484 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ¬ (vol*‘𝑥) ∈ ℝ)
32 xrge0nre 13493 . . . . . . . . . . 11 (((vol*‘𝑥) ∈ (0[,]+∞) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
3330, 31, 32syl2anc 584 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
3433eqcomd 2743 . . . . . . . . 9 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → +∞ = (vol*‘𝑥))
3527, 34breqtrd 5169 . . . . . . . 8 ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3635adantlr 715 . . . . . . 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 813 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
3837ex 412 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
3912eqcomd 2743 . . . . . . . 8 ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))))
40393adant2 1132 . . . . . . 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 1138 . . . . . . 7 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
4240, 41eqbrtrd 5165 . . . . . 6 ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
43423exp 1120 . . . . 5 (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))))
4438, 43impbid 212 . . . 4 (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
4544ralbiia 3091 . . 3 (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥𝐴)) + (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥))
4645anbi2i 623 . 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 275 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cdif 3948  cin 3950  wss 3951  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155   + caddc 11158  +∞cpnf 11292  *cxr 11294  cle 11296   +𝑒 cxad 13152  [,]cicc 13390  vol*covol 25497  volcvol 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-ovol 25499  df-vol 25500
This theorem is referenced by:  ismbl4  46008
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