Theorem ismbl4 46707

Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 25685, but here +_𝑒 is used, and the precondition (vol*‘𝑥) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

Ref	Expression
ismbl4	⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ismbl4

Step	Hyp	Ref	Expression
1		ismbl3 46700	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
2		elpwi 4569	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3		ovolcl 25637	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ ℝ*)
5	4	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ∈ ℝ*)
6		inss1 4189	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
7	6, 2	sstrid 3948	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∩ 𝐴) ⊆ ℝ)
8		ovolcl 25637	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
10	2	ssdifssd 4101	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∖ 𝐴) ⊆ ℝ)
11		ovolcl 25637	. . . . . . . . . 10 ⊢ ((𝑥 ∖ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ*)
13	9, 12	xaddcld 13322	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
14	13	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
15	2	ovolsplit 46702	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
16	15	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
17		simpr 489	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
18	5, 14, 16, 17	xrletrid 13175	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
19	18	ex 417	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
20	13	xrleidd 13172	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
21	20	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
22		id 23	. . . . . . . . 9 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
23	22	eqcomd 2769	. . . . . . . 8 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘𝑥))
24	23	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘𝑥))
25	21, 24	breqtrd 5137	. . . . . 6 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))
26	25	ex 417	. . . . 5 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
27	19, 26	impbid 215	. . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → (((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
28	27	ralbiia 3109	. . 3 ⊢ (∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥) ↔ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
29	28	anbi2i 634	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
30	1, 29	bitri 278	1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562   class class class wbr 5109  dom cdm 5661  ‘cfv 6536  (class class class)co 7410  ℝcr 11094  ℝ^*cxr 11237   ≤ cle 11239   +_𝑒 cxad 13130  vol*covol 25621  volcvol 25622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-inf 9399  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-q 12968  df-rp 13012  df-xadd 13133  df-ioo 13371  df-ico 13373  df-icc 13374  df-fz 13531  df-fl 13821  df-seq 14034  df-exp 14094  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-ovol 25623  df-vol 25624

This theorem is referenced by:  vonvolmbl  47375