vonvolmbllem - Mathbox for Glauco Siliprandi

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Theorem vonvolmbllem 46107

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 2892	. . . . . . . 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 44649	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑌 ↑_m {𝐴}))
6	5	ineq1d 4206	. . . . . 6 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
7		reex 11224	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9	3	sselda 3973	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑_m {𝐴}))
10		elmapi 8861	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m {𝐴}) → 𝑓:{𝐴}⟶ℝ)
11	9, 10	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶ℝ)
12	11	frnd 6725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ)
13	12	ralrimiva 3136	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ)
14		iunss 5044	. . . . . . . . . 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 4022	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
17	8, 16	ssexd 5320	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
18		vonvolmbllem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
19	8, 18	ssexd 5320	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
20		snex 5428	. . . . . . . 8 ⊢ {𝐴} ∈ V
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} ∈ V)
22	17, 19, 21	inmap 44642	. . . . . 6 ⊢ (𝜑 → ((𝑌 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ∩ 𝐵) ↑_m {𝐴}))
23	6, 22	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ∩ 𝐵) ↑_m {𝐴}))
24	23	fveq2d 6894	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑌 ∩ 𝐵) ↑_m {𝐴})))
25	16	ssinss1d 44473	. . . . 5 ⊢ (𝜑 → (𝑌 ∩ 𝐵) ⊆ ℝ)
26	2, 25	ovnovol 46106	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑌 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑌 ∩ 𝐵)))
27	24, 26	eqtrd 2765	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) = (vol‘(𝑌 ∩ 𝐵)))
28	5	difeq1d 4114	. . . . . 6 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
29	17, 19, 2	difmapsn 44645	. . . . . 6 ⊢ (𝜑 → ((𝑌 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ∖ 𝐵) ↑_m {𝐴}))
30	28, 29	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ∖ 𝐵) ↑_m {𝐴}))
31	30	fveq2d 6894	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑌 ∖ 𝐵) ↑_m {𝐴})))
32	16	ssdifssd 4136	. . . . 5 ⊢ (𝜑 → (𝑌 ∖ 𝐵) ⊆ ℝ)
33	2, 32	ovnovol 46106	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑌 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑌 ∖ 𝐵)))
34	31, 33	eqtrd 2765	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴}))) = (vol‘(𝑌 ∖ 𝐵)))
35	27, 34	oveq12d 7431	. 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 6894	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((voln‘{𝐴})‘(𝑌 ↑_m {𝐴})))
37	2, 16	ovnovol 46106	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑌 ↑_m {𝐴})) = (vol‘𝑌))
38	17, 16	elpwd 4605	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 ℝ)
39		vonvolmbllem.e	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵))))
40		fveq2 6890	. . . . . 6 ⊢ (𝑦 = 𝑌 → (vol‘𝑦) = (vol‘𝑌))
41		ineq1 4200	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐵) = (𝑌 ∩ 𝐵))
42	41	fveq2d 6894	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol‘(𝑦 ∩ 𝐵)) = (vol‘(𝑌 ∩ 𝐵)))
43		difeq1 4108	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∖ 𝐵) = (𝑌 ∖ 𝐵))
44	43	fveq2d 6894	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol‘(𝑦 ∖ 𝐵)) = (vol‘(𝑌 ∖ 𝐵)))
45	42, 44	oveq12d 7431	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵))))
46	40, 45	eqeq12d 2741	. . . . 5 ⊢ (𝑦 = 𝑌 → ((vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) ↔ (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵)))))
47	46	rspcva 3601	. . . 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 582	. . 3 ⊢ (𝜑 → (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol*‘(𝑌 ∖ 𝐵))))
49	36, 37, 48	3eqtrd 2769	. 2 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol*‘(𝑌 ∖ 𝐵))))
50	35, 49	eqtr4d 2768	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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ∖ cdif 3938 ∩ cin 3940 ⊆ wss 3941 𝒫 cpw 4599 {csn 4625 ∪ ciun 4992 ran crn 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 ↑_m cmap 8838 ℝcr 11132 +_𝑒 cxad 13117 vol*covol 25404 voln*covoln 45983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-rlim 15460 df-sum 15660 df-prod 15877 df-rest 17398 df-topgen 17419 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-top 22809 df-topon 22826 df-bases 22862 df-cmp 23304 df-ovol 25406 df-vol 25407 df-sumge0 45810 df-ovoln 45984

This theorem is referenced by: vonvolmbl 46108

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