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 6685 | . 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 42914 | . 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 42936 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝑦 ∈ 𝑆) |
9 | 8 | ssd 41364 | . 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ⊆ 𝑆) |
10 | eqid 2821 | . . . 4 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋) | |
11 | 3, 10 | unidmvon 42919 | . . 3 ⊢ (𝜑 → ∪ dom (voln‘𝑋) = (ℝ ↑m 𝑋)) |
12 | 4 | unieqi 4851 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom (voln‘𝑋) |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → ∪ 𝑆 = ∪ dom (voln‘𝑋)) |
14 | rrxunitopnfi 42597 | . . . 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 2866 | . 2 ⊢ (𝜑 → ∪ 𝑆 = ∪ (TopOpen‘(ℝ^‘𝑋))) |
17 | 1, 2, 5, 9, 16 | salgenss 42639 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∪ cuni 4838 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 Fincfn 8509 ℝcr 10536 TopOpenctopn 16695 ℝ^crrx 23986 SalGencsalgen 42617 volncvoln 42840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-prod 15260 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-field 19505 df-subrg 19533 df-abv 19588 df-staf 19616 df-srng 19617 df-lmod 19636 df-lss 19704 df-lmhm 19794 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-refld 20749 df-phl 20770 df-dsmm 20876 df-frlm 20891 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cmp 21995 df-xms 22930 df-ms 22931 df-nm 23192 df-ngp 23193 df-tng 23194 df-nrg 23195 df-nlm 23196 df-clm 23667 df-cph 23772 df-tcph 23773 df-rrx 23988 df-ovol 24065 df-vol 24066 df-salg 42614 df-salgen 42618 df-sumge0 42665 df-mea 42752 df-ome 42792 df-caragen 42794 df-ovoln 42839 df-voln 42841 |
This theorem is referenced by: bormflebmf 43050 |
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