Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 6732 | . 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 43825 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑋 ∈ Fin) |
7 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) | |
8 | 6, 4, 7 | opnvonmbl 43847 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ 𝑆) |
9 | 8 | ssd 42303 | . 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ⊆ 𝑆) |
10 | eqid 2737 | . . . 4 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
11 | 3, 10 | unidmvon 43830 | . . 3 ⊢ (𝜑 → ∪ dom (voln‘𝑋) = (ℝ ↑m 𝑋)) |
12 | 4 | unieqi 4832 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom (voln‘𝑋) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom (voln‘𝑋)) |
14 | rrxunitopnfi 43508 | . . . 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 2787 | . 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ (TopOpen‘(ℝ^‘𝑋))) |
17 | 1, 2, 5, 9, 16 | salgenss 43550 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 ∪ cuni 4819 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 Fincfn 8626 ℝcr 10728 TopOpenctopn 16926 ℝ^crrx 24280 SalGencsalgen 43528 volncvoln 43751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-ac2 10077 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-ac 9730 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-prod 15468 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-prds 16952 df-pws 16954 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-ghm 18620 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-rnghom 19735 df-drng 19769 df-field 19770 df-subrg 19798 df-abv 19853 df-staf 19881 df-srng 19882 df-lmod 19901 df-lss 19969 df-lmhm 20059 df-lvec 20140 df-sra 20209 df-rgmod 20210 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-refld 20567 df-phl 20588 df-dsmm 20694 df-frlm 20709 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cmp 22284 df-xms 23218 df-ms 23219 df-nm 23480 df-ngp 23481 df-tng 23482 df-nrg 23483 df-nlm 23484 df-clm 23960 df-cph 24065 df-tcph 24066 df-rrx 24282 df-ovol 24361 df-vol 24362 df-salg 43525 df-salgen 43529 df-sumge0 43576 df-mea 43663 df-ome 43703 df-caragen 43705 df-ovoln 43750 df-voln 43752 |
This theorem is referenced by: bormflebmf 43961 |
Copyright terms: Public domain | W3C validator |