Theorem ismbl3 45907

Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 25581, 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*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl3

Step	Hyp	Ref	Expression
1		ismbl2 25581	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
2		inss1 4258	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
3	2	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ 𝐴) ⊆ 𝑥)
4		elpwi 4629	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5	4	adantr 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 25540	. . . . . . . . . . 11 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
8	3, 5, 6, 7	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
9		difssd 4160	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ 𝐴) ⊆ 𝑥)
10		ovolsscl 25540	. . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
11	9, 5, 6, 10	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
12	8, 11	rexaddd 13296	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
13	12	adantlr 714	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
14		id 22	. . . . . . . . . 10 ⊢ (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
15	14	imp 406	. . . . . . . . 9 ⊢ ((((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
16	15	adantll 713	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
17	13, 16	eqbrtrd 5188	. . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
18	2, 4	sstrid 4020	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∩ 𝐴) ⊆ ℝ)
19		ovolcl 25532	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
20	18, 19	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
21	4	ssdifssd 4170	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∖ 𝐴) ⊆ ℝ)
22		ovolcl 25532	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
24	20, 23	xaddcld 13363	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
25		pnfge 13193	. . . . . . . . . . 11 ⊢ (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ* → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ +∞)
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ +∞)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ +∞)
28		ovolf 25536	. . . . . . . . . . . . 13 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
29	28	ffvelcdmi 7117	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞))
30	29	adantr 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 13513	. . . . . . . . . . 11 ⊢ (((vol*‘𝑥) ∈ (0[,]+∞) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
33	30, 31, 32	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
34	33	eqcomd 2746	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → +∞ = (vol*‘𝑥))
35	27, 34	breqtrd 5192	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
36	35	adantlr 714	. . . . . . 7 ⊢ (((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
37	17, 36	pm2.61dan 812	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
38	37	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
39	12	eqcomd 2746	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
40	39	3adant2 1131	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
41		simp2 1137	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
42	40, 41	eqbrtrd 5188	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
43	42	3exp 1119	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
44	38, 43	impbid 212	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) ↔ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
45	44	ralbiia 3097	. . 3 ⊢ (∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
46	45	anbi2i 622	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
47	1, 46	bitri 275	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184   + caddc 11187  +∞cpnf 11321  ℝ^*cxr 11323   ≤ cle 11325   +_𝑒 cxad 13173  [,]cicc 13410  vol*covol 25516  volcvol 25517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xadd 13176  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fl 13843  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-ovol 25518  df-vol 25519

This theorem is referenced by:  ismbl4  45914