Theorem ismbl3 46758

Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 25737, 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 25737	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))))
2		inss1 4189	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
3	2	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ 𝐴) ⊆ 𝑥)
4		elpwi 4571	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
5	4	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆ ℝ)
6		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ ℝ)
7		ovolsscl 25696	. . . . . . . . . . 11 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
8	3, 5, 6, 7	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ)
9		difssd 4091	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ 𝐴) ⊆ 𝑥)
10		ovolsscl 25696	. . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
11	9, 5, 6, 10	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ)
12	8, 11	rexaddd 13276	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))
13	12	adantlr 728	. . . . . . . 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 23	. . . . . . . . . 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 412	. . . . . . . . 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 727	. . . . . . . 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 5135	. . . . . . 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 3949	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∩ 𝐴) ⊆ ℝ)
19		ovolcl 25688	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
20	18, 19	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
21	4	ssdifssd 4101	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∖ 𝐴) ⊆ ℝ)
22		ovolcl 25688	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∖ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
23	21, 22	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
24	20, 23	xaddcld 13343	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
25		pnfge 13171	. . . . . . . . . . 11 ⊢ (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ* → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ +∞)
26	24, 25	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ +∞)
27	26	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ +∞)
28		ovolf 25692	. . . . . . . . . . . . 13 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
29	28	ffvelcdmi 7082	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞))
30	29	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) ∈ (0[,]+∞))
31		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ¬ (vol*‘𝑥) ∈ ℝ)
32		xrge0nre 13496	. . . . . . . . . . 11 ⊢ (((vol*‘𝑥) ∈ (0[,]+∞) ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
33	30, 31, 32	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = +∞)
34	33	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → +∞ = (vol*‘𝑥))
35	27, 34	breqtrd 5139	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ¬ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
36	35	adantlr 728	. . . . . . 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 825	. . . . . 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 418	. . . . 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 2771	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
40	39	3adant2 1149	. . . . . . 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 1155	. . . . . . 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 5135	. . . . . 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 1137	. . . . 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 215	. . . 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 3111	. . 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 635	. 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 278	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564   class class class wbr 5111  dom cdm 5663  ‘cfv 6540  (class class class)co 7419  ℝcr 11114  0cc0 11115   + caddc 11118  +∞cpnf 11255  ℝ^*cxr 11257   ≤ cle 11259   +_𝑒 cxad 13151  [,]cicc 13391  vol*covol 25672  volcvol 25673

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-sup 9409  df-inf 9410  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-z 12607  df-uz 12879  df-q 12989  df-rp 13033  df-xadd 13154  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13552  df-fl 13843  df-seq 14056  df-exp 14116  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-ovol 25674  df-vol 25675

This theorem is used by:  ismbl4  46765