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 6688 | . 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 42901 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑋 ∈ Fin) |
7 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) | |
8 | 6, 4, 7 | opnvonmbl 42923 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ 𝑆) |
9 | 8 | ssd 41350 | . 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ⊆ 𝑆) |
10 | eqid 2824 | . . . 4 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
11 | 3, 10 | unidmvon 42906 | . . 3 ⊢ (𝜑 → ∪ dom (voln‘𝑋) = (ℝ ↑m 𝑋)) |
12 | 4 | unieqi 4854 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom (voln‘𝑋) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom (voln‘𝑋)) |
14 | rrxunitopnfi 42584 | . . . 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 2869 | . 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ (TopOpen‘(ℝ^‘𝑋))) |
17 | 1, 2, 5, 9, 16 | salgenss 42626 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 ∪ cuni 4841 dom cdm 5558 ‘cfv 6358 (class class class)co 7159 ↑m cmap 8409 Fincfn 8512 ℝcr 10539 TopOpenctopn 16698 ℝ^crrx 23989 SalGencsalgen 42604 volncvoln 42827 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cc 9860 ax-ac2 9888 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-disj 5035 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-dju 9333 df-card 9371 df-acn 9374 df-ac 9545 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-rlim 14849 df-sum 15046 df-prod 15263 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-prds 16724 df-pws 16726 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-field 19508 df-subrg 19536 df-abv 19591 df-staf 19619 df-srng 19620 df-lmod 19639 df-lss 19707 df-lmhm 19797 df-lvec 19878 df-sra 19947 df-rgmod 19948 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-refld 20752 df-phl 20773 df-dsmm 20879 df-frlm 20894 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cmp 21998 df-xms 22933 df-ms 22934 df-nm 23195 df-ngp 23196 df-tng 23197 df-nrg 23198 df-nlm 23199 df-clm 23670 df-cph 23775 df-tcph 23776 df-rrx 23991 df-ovol 24068 df-vol 24069 df-salg 42601 df-salgen 42605 df-sumge0 42652 df-mea 42739 df-ome 42779 df-caragen 42781 df-ovoln 42826 df-voln 42828 |
This theorem is referenced by: bormflebmf 43037 |
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