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Mirrors > Home > MPE Home > Th. List > nulmbl | Structured version Visualization version GIF version |
Description: A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
Ref | Expression |
---|---|
nulmbl | ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 475 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ⊆ ℝ) | |
2 | elpwi 4359 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
3 | inss2 4029 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
4 | ovolssnul 23595 | . . . . . . . . . 10 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝑥 ∩ 𝐴)) = 0) | |
5 | 3, 4 | mp3an1 1573 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝑥 ∩ 𝐴)) = 0) |
6 | 5 | adantr 473 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) = 0) |
7 | 6 | oveq1d 6893 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = (0 + (vol*‘(𝑥 ∖ 𝐴)))) |
8 | difss 3935 | . . . . . . . . . . 11 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥 | |
9 | ovolsscl 23594 | . . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) | |
10 | 8, 9 | mp3an1 1573 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) |
11 | 10 | adantl 474 | . . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) |
12 | 11 | recnd 10357 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℂ) |
13 | 12 | addid2d 10527 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (0 + (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘(𝑥 ∖ 𝐴))) |
14 | 7, 13 | eqtrd 2833 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘(𝑥 ∖ 𝐴))) |
15 | simprl 788 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ) | |
16 | ovolss 23593 | . . . . . . 7 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ≤ (vol*‘𝑥)) | |
17 | 8, 15, 16 | sylancr 582 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ≤ (vol*‘𝑥)) |
18 | 14, 17 | eqbrtrd 4865 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) |
19 | 18 | expr 449 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
20 | 2, 19 | sylan2 587 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
21 | 20 | ralrimiva 3147 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
22 | ismbl2 23635 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))) | |
23 | 1, 21, 22 | sylanbrc 579 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∖ cdif 3766 ∩ cin 3768 ⊆ wss 3769 𝒫 cpw 4349 class class class wbr 4843 dom cdm 5312 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 + caddc 10227 ≤ cle 10364 vol*covol 23570 volcvol 23571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fl 12848 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-ovol 23572 df-vol 23573 |
This theorem is referenced by: 0mbl 23647 icombl1 23671 ioombl 23673 ovolioo 23676 uniiccmbl 23698 volivth 23715 mbfeqalem1 23749 itg10a 23818 itg2uba 23851 itgss3 23922 cntnevol 30807 voliunnfl 33942 volsupnfl 33943 cnambfre 33946 snmbl 40918 |
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