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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ctvonmbl | Structured version Visualization version GIF version |
Description: Any n-dimensional countable set is Lebesgue measurable. This is the second statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
ctvonmbl.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
ctvonmbl.2 | ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋)) |
ctvonmbl.3 | ⊢ (𝜑 → 𝐴 ≼ ω) |
Ref | Expression |
---|---|
ctvonmbl | ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunid 4607 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
2 | ctvonmbl.1 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
3 | 2 | vonmea 41109 | . . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas) |
4 | eqid 2651 | . . . 4 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
5 | 3, 4 | dmmeasal 40987 | . . 3 ⊢ (𝜑 → dom (voln‘𝑋) ∈ SAlg) |
6 | ctvonmbl.3 | . . 3 ⊢ (𝜑 → 𝐴 ≼ ω) | |
7 | 2 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ Fin) |
8 | ctvonmbl.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋)) | |
9 | 8 | sselda 3636 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (ℝ ↑𝑚 𝑋)) |
10 | 7, 9 | snvonmbl 41221 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ dom (voln‘𝑋)) |
11 | 5, 6, 10 | saliuncl 40860 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 {𝑥} ∈ dom (voln‘𝑋)) |
12 | 1, 11 | syl5eqelr 2735 | 1 ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2030 ⊆ wss 3607 {csn 4210 ∪ ciun 4552 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 ωcom 7107 ↑𝑚 cmap 7899 ≼ cdom 7995 Fincfn 7997 ℝcr 9973 volncvoln 41073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cc 9295 ax-ac2 9323 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-ac 8977 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-rlim 14264 df-sum 14461 df-prod 14680 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-prds 16155 df-pws 16157 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-ghm 17705 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-rnghom 18763 df-drng 18797 df-field 18798 df-subrg 18826 df-abv 18865 df-staf 18893 df-srng 18894 df-lmod 18913 df-lss 18981 df-lmhm 19070 df-lvec 19151 df-sra 19220 df-rgmod 19221 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-refld 19999 df-phl 20019 df-dsmm 20124 df-frlm 20139 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cmp 21238 df-xms 22172 df-ms 22173 df-nm 22434 df-ngp 22435 df-tng 22436 df-nrg 22437 df-nlm 22438 df-clm 22909 df-cph 23014 df-tch 23015 df-rrx 23219 df-ovol 23279 df-vol 23280 df-salg 40847 df-sumge0 40898 df-mea 40985 df-ome 41025 df-caragen 41027 df-ovoln 41072 df-voln 41074 |
This theorem is referenced by: (None) |
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