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Mirrors > Home > MPE Home > Th. List > mblss | Structured version Visualization version GIF version |
Description: A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.) |
Ref | Expression |
---|---|
mblss | ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbl 24690 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 𝒫 cpw 4533 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 + caddc 10874 vol*covol 24626 volcvol 24627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ico 13085 df-icc 13086 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-ovol 24628 df-vol 24629 |
This theorem is referenced by: volss 24697 nulmbl2 24700 unmbl 24701 shftmbl 24702 unidmvol 24705 inmbl 24706 difmbl 24707 volun 24709 volinun 24710 volfiniun 24711 voliunlem2 24715 voliunlem3 24716 volsup 24720 volsup2 24769 volcn 24770 vitalilem4 24775 vitalilem5 24776 vitali 24777 ismbf 24792 ismbfcn 24793 mbfconst 24797 mbfid 24799 cncombf 24822 cnmbf 24823 i1fima2 24843 i1fd 24845 itg1ge0 24850 i1f1lem 24853 itg11 24855 i1fadd 24859 i1fmul 24860 itg1addlem2 24861 itg1addlem5 24865 i1fres 24870 itg1ge0a 24876 itg1climres 24879 mbfi1fseqlem4 24883 mbfi1flim 24888 mbfmullem2 24889 itg2const2 24906 itg2splitlem 24913 itg2split 24914 itg2gt0 24925 itg2cnlem2 24927 ibladdlem 24984 itgaddlem1 24987 iblabslem 24992 itggt0 25008 itgcn 25009 ftc1lem4 25203 itgulm 25567 areaf 26111 dmvlsiga 32097 volsupnfl 35822 cnambfre 35825 itg2addnclem 35828 ibladdnclem 35833 itgaddnclem1 35835 iblabsnclem 35840 ftc1cnnclem 35848 volge0 43502 dmvolss 43526 vonvol 44200 |
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