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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 24290 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∖ cdif 3850 ∩ cin 3852 ⊆ wss 3853 𝒫 cpw 4498 dom cdm 5535 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 + caddc 10630 vol*covol 24226 volcvol 24227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-ico 12839 df-icc 12840 df-fz 12994 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-ovol 24228 df-vol 24229 |
This theorem is referenced by: volss 24297 nulmbl2 24300 unmbl 24301 shftmbl 24302 unidmvol 24305 inmbl 24306 difmbl 24307 volun 24309 volinun 24310 volfiniun 24311 voliunlem2 24315 voliunlem3 24316 volsup 24320 volsup2 24369 volcn 24370 vitalilem4 24375 vitalilem5 24376 vitali 24377 ismbf 24392 ismbfcn 24393 mbfconst 24397 mbfid 24399 cncombf 24422 cnmbf 24423 i1fima2 24443 i1fd 24445 itg1ge0 24450 i1f1lem 24453 itg11 24455 i1fadd 24459 i1fmul 24460 itg1addlem2 24461 itg1addlem5 24465 i1fres 24470 itg1ge0a 24476 itg1climres 24479 mbfi1fseqlem4 24483 mbfi1flim 24488 mbfmullem2 24489 itg2const2 24506 itg2splitlem 24513 itg2split 24514 itg2gt0 24525 itg2cnlem2 24527 ibladdlem 24584 itgaddlem1 24587 iblabslem 24592 itggt0 24608 itgcn 24609 ftc1lem4 24803 itgulm 25167 areaf 25711 dmvlsiga 31679 volsupnfl 35477 cnambfre 35480 itg2addnclem 35483 ibladdnclem 35488 itgaddnclem1 35490 iblabsnclem 35495 ftc1cnnclem 35503 volge0 43084 dmvolss 43108 vonvol 43782 |
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