vonvolmbllem - Mathbox for Glauco Siliprandi

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Theorem vonvolmbllem 45961

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 2898	. . . . . . . 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 44502	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑌 ↑_m {𝐴}))
6	5	ineq1d 4207	. . . . . 6 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
7		reex 11215	. . . . . . . . 9 ⊢ ℝ ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
9	3	sselda 3978	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑_m {𝐴}))
10		elmapi 8857	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑_m {𝐴}) → 𝑓:{𝐴}⟶ℝ)
11	9, 10	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶ℝ)
12	11	frnd 6724	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ)
13	12	ralrimiva 3141	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ)
14		iunss 5042	. . . . . . . . . 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 4026	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
17	8, 16	ssexd 5318	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
18		vonvolmbllem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
19	8, 18	ssexd 5318	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
20		snex 5427	. . . . . . . 8 ⊢ {𝐴} ∈ V
21	20	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} ∈ V)
22	17, 19, 21	inmap 44495	. . . . . 6 ⊢ (𝜑 → ((𝑌 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ∩ 𝐵) ↑_m {𝐴}))
23	6, 22	eqtrd 2767	. . . . 5 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑_m {𝐴})) = ((𝑌 ∩ 𝐵) ↑_m {𝐴}))
24	23	fveq2d 6895	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑌 ∩ 𝐵) ↑_m {𝐴})))
25	16	ssinss1d 44325	. . . . 5 ⊢ (𝜑 → (𝑌 ∩ 𝐵) ⊆ ℝ)
26	2, 25	ovnovol 45960	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑌 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑌 ∩ 𝐵)))
27	24, 26	eqtrd 2767	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∩ (𝐵 ↑_m {𝐴}))) = (vol‘(𝑌 ∩ 𝐵)))
28	5	difeq1d 4117	. . . . . 6 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
29	17, 19, 2	difmapsn 44498	. . . . . 6 ⊢ (𝜑 → ((𝑌 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ∖ 𝐵) ↑_m {𝐴}))
30	28, 29	eqtrd 2767	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑_m {𝐴})) = ((𝑌 ∖ 𝐵) ↑_m {𝐴}))
31	30	fveq2d 6895	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑌 ∖ 𝐵) ↑_m {𝐴})))
32	16	ssdifssd 4138	. . . . 5 ⊢ (𝜑 → (𝑌 ∖ 𝐵) ⊆ ℝ)
33	2, 32	ovnovol 45960	. . . 4 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑌 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑌 ∖ 𝐵)))
34	31, 33	eqtrd 2767	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑋 ∖ (𝐵 ↑_m {𝐴}))) = (vol‘(𝑌 ∖ 𝐵)))
35	27, 34	oveq12d 7432	. 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 6895	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((voln‘{𝐴})‘(𝑌 ↑_m {𝐴})))
37	2, 16	ovnovol 45960	. . 3 ⊢ (𝜑 → ((voln‘{𝐴})‘(𝑌 ↑_m {𝐴})) = (vol‘𝑌))
38	17, 16	elpwd 4604	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 ℝ)
39		vonvolmbllem.e	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵))))
40		fveq2 6891	. . . . . 6 ⊢ (𝑦 = 𝑌 → (vol‘𝑦) = (vol‘𝑌))
41		ineq1 4201	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐵) = (𝑌 ∩ 𝐵))
42	41	fveq2d 6895	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol‘(𝑦 ∩ 𝐵)) = (vol‘(𝑌 ∩ 𝐵)))
43		difeq1 4111	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∖ 𝐵) = (𝑌 ∖ 𝐵))
44	43	fveq2d 6895	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol‘(𝑦 ∖ 𝐵)) = (vol‘(𝑌 ∖ 𝐵)))
45	42, 44	oveq12d 7432	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵))))
46	40, 45	eqeq12d 2743	. . . . 5 ⊢ (𝑦 = 𝑌 → ((vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) ↔ (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol‘(𝑌 ∖ 𝐵)))))
47	46	rspcva 3605	. . . 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 583	. . 3 ⊢ (𝜑 → (vol‘𝑌) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol*‘(𝑌 ∖ 𝐵))))
49	36, 37, 48	3eqtrd 2771	. 2 ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = ((vol‘(𝑌 ∩ 𝐵)) +_𝑒 (vol*‘(𝑌 ∖ 𝐵))))
50	35, 49	eqtr4d 2770	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 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 𝒫 cpw 4598 {csn 4624 ∪ ciun 4991 ran crn 5673 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8834 ℝcr 11123 +_𝑒 cxad 13108 vol*covol 25365 voln*covoln 45837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-rlim 15451 df-sum 15651 df-prod 15868 df-rest 17389 df-topgen 17410 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-top 22770 df-topon 22787 df-bases 22823 df-cmp 23265 df-ovol 25367 df-vol 25368 df-sumge0 45664 df-ovoln 45838

This theorem is referenced by: vonvolmbl 45962

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