rrnmbl - Mathbox for Glauco Siliprandi

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Theorem rrnmbl 45628

Description: The set of n-dimensional Real numbers is Lebesgue measurable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypothesis**
Ref	Expression
rrnmbl.1	⊢ (𝜑 → 𝑋 ∈ Fin)

**Assertion**
Ref	Expression
rrnmbl	⊢ (𝜑 → (ℝ ↑_m 𝑋) ∈ dom (voln‘𝑋))

**Proof of Theorem rrnmbl**
Step	Hyp	Ref	Expression
1		rrnmbl.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
2	1	ovnome 45587	. . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas)
3		eqid 2730	. . 3 ⊢ ∪ dom (voln‘𝑋) = ∪ dom (voln‘𝑋)
4		eqid 2730	. . 3 ⊢ (CaraGen‘(voln‘𝑋)) = (CaraGen‘(voln‘𝑋))
5	2, 3, 4	caragenunidm 45522	. 2 ⊢ (𝜑 → ∪ dom (voln‘𝑋) ∈ (CaraGen‘(voln‘𝑋)))
6	1	dmovn 45618	. . . . 5 ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑_m 𝑋))
7	6	unieqd 4921	. . . 4 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = ∪ 𝒫 (ℝ ↑_m 𝑋))
8		unipw 5449	. . . . 5 ⊢ ∪ 𝒫 (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋)
9	8	a1i 11	. . . 4 ⊢ (𝜑 → ∪ 𝒫 (ℝ ↑_m 𝑋) = (ℝ ↑_m 𝑋))
10	7, 9	eqtr2d 2771	. . 3 ⊢ (𝜑 → (ℝ ↑_m 𝑋) = ∪ dom (voln*‘𝑋))
11	1	dmvon 45620	. . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
12	10, 11	eleq12d 2825	. 2 ⊢ (𝜑 → ((ℝ ↑_m 𝑋) ∈ dom (voln‘𝑋) ↔ ∪ dom (voln‘𝑋) ∈ (CaraGen‘(voln‘𝑋))))
13	5, 12	mpbird 256	1 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ∈ dom (voln‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 𝒫 cpw 4601 ∪ cuni 4907 dom cdm 5675 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 Fincfn 8941 ℝcr 11111 CaraGenccaragen 45505 voln*covoln 45550 volncvoln 45552

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 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 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-prod 15854 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-rest 17372 df-0g 17391 df-topgen 17393 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-cnfld 21145 df-top 22616 df-topon 22633 df-bases 22669 df-cmp 23111 df-ovol 25213 df-vol 25214 df-sumge0 45377 df-ome 45504 df-caragen 45506 df-ovoln 45551 df-voln 45553

This theorem is referenced by: hoimbl 45645

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