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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > borelmbl | Structured version Visualization version GIF version |
Description: All Borel subsets of the n-dimensional Real numbers are Lebesgue measurable. This is Proposition 115G (b) of [Fremlin1] p. 32. See also Definition 111G (d) of [Fremlin1] p. 13. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
borelmbl.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
borelmbl.s | ⊢ 𝑆 = dom (voln‘𝑋) |
borelmbl.b | ⊢ 𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋))) |
Ref | Expression |
---|---|
borelmbl | ⊢ (𝜑 → 𝐵 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6364 | . 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ∈ V) | |
2 | borelmbl.b | . 2 ⊢ 𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋))) | |
3 | borelmbl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
4 | borelmbl.s | . . 3 ⊢ 𝑆 = dom (voln‘𝑋) | |
5 | 3, 4 | dmovnsal 41332 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | 3 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑋 ∈ Fin) |
7 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) | |
8 | 6, 4, 7 | opnvonmbl 41354 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ 𝑆) |
9 | 8 | ssd 39751 | . 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ⊆ 𝑆) |
10 | eqid 2760 | . . . 4 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
11 | 3, 10 | unidmvon 41337 | . . 3 ⊢ (𝜑 → ∪ dom (voln‘𝑋) = (ℝ ↑𝑚 𝑋)) |
12 | 4 | unieqi 4597 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom (voln‘𝑋) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom (voln‘𝑋)) |
14 | rrxunitopnfi 41015 | . . . 4 ⊢ (𝑋 ∈ Fin → ∪ (TopOpen‘(ℝ^‘𝑋)) = (ℝ ↑𝑚 𝑋)) | |
15 | 3, 14 | syl 17 | . . 3 ⊢ (𝜑 → ∪ (TopOpen‘(ℝ^‘𝑋)) = (ℝ ↑𝑚 𝑋)) |
16 | 11, 13, 15 | 3eqtr4d 2804 | . 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ (TopOpen‘(ℝ^‘𝑋))) |
17 | 1, 2, 5, 9, 16 | salgenss 41057 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 ∪ cuni 4588 dom cdm 5266 ‘cfv 6049 (class class class)co 6813 ↑𝑚 cmap 8023 Fincfn 8121 ℝcr 10127 TopOpenctopn 16284 ℝ^crrx 23371 SalGencsalgen 41035 volncvoln 41258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cc 9449 ax-ac2 9477 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-disj 4773 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-tpos 7521 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-acn 8958 df-ac 9129 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-rlim 14419 df-sum 14616 df-prod 14835 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-prds 16310 df-pws 16312 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-ghm 17859 df-cntz 17950 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-cring 18750 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-rnghom 18917 df-drng 18951 df-field 18952 df-subrg 18980 df-abv 19019 df-staf 19047 df-srng 19048 df-lmod 19067 df-lss 19135 df-lmhm 19224 df-lvec 19305 df-sra 19374 df-rgmod 19375 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-cnfld 19949 df-refld 20153 df-phl 20173 df-dsmm 20278 df-frlm 20293 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cmp 21392 df-xms 22326 df-ms 22327 df-nm 22588 df-ngp 22589 df-tng 22590 df-nrg 22591 df-nlm 22592 df-clm 23063 df-cph 23168 df-tch 23169 df-rrx 23373 df-ovol 23433 df-vol 23434 df-salg 41032 df-salgen 41036 df-sumge0 41083 df-mea 41170 df-ome 41210 df-caragen 41212 df-ovoln 41257 df-voln 41259 |
This theorem is referenced by: bormflebmf 41468 |
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