vonvolmbllem - Mathbox for Glauco Siliprandi

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Theorem vonvolmbllem 45362

Description: If a subset 𝐵 of real numbers is Lebesgue measurable, then its corresponding 1-dimensional set is measurable w.r.t. the n-dimensional Lebesgue measure, (with 𝑛 equal to 1). (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
vonvolmbllem.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
vonvolmbllem.b	⊢ (𝜑 → 𝐵 ⊆ ℝ)
vonvolmbllem.e	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵))))
vonvolmbllem.x	⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑_m {𝐴}))
vonvolmbllem.y	⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓

**Assertion**
Ref	Expression
vonvolmbllem	⊢ (𝜑 → (((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴})))) = ((voln*‘{𝐴})‘𝑋))

Distinct variable groups: 𝐴,𝑓 𝑦,𝐵 𝑓,𝑋 𝑓,𝑌 𝑦,𝑌 𝜑,𝑓 Allowed substitution hints: 𝜑(𝑦) 𝐴(𝑦) 𝐵(𝑓) 𝑉(𝑦,𝑓) 𝑋(𝑦)

**Proof of Theorem vonvolmbllem**
Step	Hyp	Ref	Expression
1		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑓𝑌
2		vonvolmbllem.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		vonvolmbllem.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑_m {𝐴}))
4		vonvolmbllem.y	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
5	1, 2, 3, 4	ssmapsn 43900	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑌 ↑_m {𝐴}))
6	5	ineq1d 4210	. . . . . 6 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
7		reex 11197	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9	3	sselda 3981	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑_m {𝐴}))
10		elmapi 8839	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m {𝐴}) → 𝑓:{𝐴}⟶ℝ)
11	9, 10	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶ℝ)
12	11	frnd 6722	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ)
13	12	ralrimiva 3146	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ)
14		iunss 5047	. . . . . . . . . 10 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ)
15	13, 14	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ)
16	4, 15	eqsstrid 4029	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
17	8, 16	ssexd 5323	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
18		vonvolmbllem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
19	8, 18	ssexd 5323	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
20		snex 5430	. . . . . . . 8 ⊢ {𝐴} ∈ V
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} ∈ V)
22	17, 19, 21	inmap 43893	. . . . . 6 ⊢ (𝜑 → ((𝑌 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ∩ 𝐵) ↑_m {𝐴}))
23	6, 22	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ∩ 𝐵) ↑_m {𝐴}))
24	23	fveq2d 6892	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑌 ∩ 𝐵) ↑_m {𝐴})))
25	16	ssinss1d 43720	. . . . 5 ⊢ (𝜑 → (𝑌 ∩ 𝐵) ⊆ ℝ)
26	2, 25	ovnovol 45361	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑌 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑌 ∩ 𝐵)))
27	24, 26	eqtrd 2772	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) = (vol‘(𝑌 ∩ 𝐵)))
28	5	difeq1d 4120	. . . . . 6 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
29	17, 19, 2	difmapsn 43896	. . . . . 6 ⊢ (𝜑 → ((𝑌 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ∖ 𝐵) ↑_m {𝐴}))
30	28, 29	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ∖ 𝐵) ↑_m {𝐴}))
31	30	fveq2d 6892	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑌 ∖ 𝐵) ↑_m {𝐴})))
32	16	ssdifssd 4141	. . . . 5 ⊢ (𝜑 → (𝑌 ∖ 𝐵) ⊆ ℝ)
33	2, 32	ovnovol 45361	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑌 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑌 ∖ 𝐵)))
34	31, 33	eqtrd 2772	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴}))) = (vol‘(𝑌 ∖ 𝐵)))
35	27, 34	oveq12d 7423	. 2 ⊢ (𝜑 → (((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴})))) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵))))
36	5	fveq2d 6892	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((voln‘{𝐴})‘(𝑌 ↑_m {𝐴})))
37	2, 16	ovnovol 45361	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑌 ↑_m {𝐴})) = (vol‘𝑌))
38	17, 16	elpwd 4607	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 ℝ)
39		vonvolmbllem.e	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵))))
40		fveq2 6888	. . . . . 6 ⊢ (𝑦 = 𝑌 → (vol‘𝑦) = (vol‘𝑌))
41		ineq1 4204	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐵) = (𝑌 ∩ 𝐵))
42	41	fveq2d 6892	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol‘(𝑦 ∩ 𝐵)) = (vol‘(𝑌 ∩ 𝐵)))
43		difeq1 4114	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∖ 𝐵) = (𝑌 ∖ 𝐵))
44	43	fveq2d 6892	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol‘(𝑦 ∖ 𝐵)) = (vol‘(𝑌 ∖ 𝐵)))
45	42, 44	oveq12d 7423	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵))))
46	40, 45	eqeq12d 2748	. . . . 5 ⊢ (𝑦 = 𝑌 → ((vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) ↔ (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵)))))
47	46	rspcva 3610	. . . 4 ⊢ ((𝑌 ∈ 𝒫 ℝ ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵)))) → (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵))))
48	38, 39, 47	syl2anc 584	. . 3 ⊢ (𝜑 → (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol*‘(𝑌 ∖ 𝐵))))
49	36, 37, 48	3eqtrd 2776	. 2 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol*‘(𝑌 ∖ 𝐵))))
50	35, 49	eqtr4d 2775	1 ⊢ (𝜑 → (((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴})))) = ((voln*‘{𝐴})‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 {csn 4627 ∪ ciun 4996 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 ℝcr 11105 +_𝑒 cxad 13086 vol*covol 24970 voln*covoln 45238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-prod 15846 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-bases 22440 df-cmp 22882 df-ovol 24972 df-vol 24973 df-sumge0 45065 df-ovoln 45239

This theorem is referenced by: vonvolmbl 45363

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