Theorem ismbl4 46822

Description: The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 25755, 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 46815	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
2		elpwi 4564	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ)
3		ovolcl 25707	. . . . . . . . 9 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ ℝ*)
5	4	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ∈ ℝ*)
6		inss1 4182	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
7	6, 2	sstrid 3942	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∩ 𝐴) ⊆ ℝ)
8		ovolcl 25707	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝐴) ⊆ ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ*)
10	2	ssdifssd 4094	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ → (𝑥 ∖ 𝐴) ⊆ ℝ)
11		ovolcl 25707	. . . . . . . . . 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 13354	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
14	13	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ∈ ℝ*)
15	2	ovolsplit 46817	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
16	15	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 ℝ ∧ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) → (vol*‘𝑥) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
17		simpr 490	. . . . . . 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 13207	. . . . . 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 418	. . . . 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 13204	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ℝ → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))))
21	20	adantr 486	. . . . . . 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 2766	. . . . . . . 8 ⊢ ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) → ((vol*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘𝑥))
24	23	adantl 487	. . . . . . 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 5131	. . . . . 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 418	. . . . 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 3106	. . 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 635	. 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)))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103  dom cdm 5655  ‘cfv 6533  (class class class)co 7414  ℝcr 11124  ℝ^*cxr 11267   ≤ cle 11269   +_𝑒 cxad 13162  vol*covol 25691  volcvol 25692

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202  ax-pre-sup 11203

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8697  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-sup 9413  df-inf 9414  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-div 11897  df-nn 12259  df-2 12328  df-3 12329  df-n0 12530  df-z 12617  df-uz 12889  df-q 12999  df-rp 13044  df-xadd 13165  df-ioo 13403  df-ico 13405  df-icc 13406  df-fz 13563  df-fl 13854  df-seq 14067  df-exp 14127  df-cj 15187  df-re 15188  df-im 15189  df-sqrt 15323  df-abs 15324  df-ovol 25693  df-vol 25694

This theorem is used by:  vonvolmbl  47490