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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 483 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ⊆ ℝ) | |
2 | elpwi 4457 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
3 | inss2 4121 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
4 | ovolssnul 23759 | . . . . . . . . . 10 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝑥 ∩ 𝐴)) = 0) | |
5 | 3, 4 | mp3an1 1438 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝑥 ∩ 𝐴)) = 0) |
6 | 5 | adantr 481 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) = 0) |
7 | 6 | oveq1d 7022 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = (0 + (vol*‘(𝑥 ∖ 𝐴)))) |
8 | difss 4024 | . . . . . . . . . . 11 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥 | |
9 | ovolsscl 23758 | . . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) | |
10 | 8, 9 | mp3an1 1438 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) |
11 | 10 | adantl 482 | . . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) |
12 | 11 | recnd 10504 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℂ) |
13 | 12 | addid2d 10677 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (0 + (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘(𝑥 ∖ 𝐴))) |
14 | 7, 13 | eqtrd 2829 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘(𝑥 ∖ 𝐴))) |
15 | simprl 767 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ) | |
16 | ovolss 23757 | . . . . . . 7 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ≤ (vol*‘𝑥)) | |
17 | 8, 15, 16 | sylancr 587 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ≤ (vol*‘𝑥)) |
18 | 14, 17 | eqbrtrd 4978 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) |
19 | 18 | expr 457 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
20 | 2, 19 | sylan2 592 | . . 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 23799 | . 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 583 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∀wral 3103 ∖ cdif 3851 ∩ cin 3853 ⊆ wss 3854 𝒫 cpw 4447 class class class wbr 4956 dom cdm 5435 ‘cfv 6217 (class class class)co 7007 ℝcr 10371 0cc0 10372 + caddc 10375 ≤ cle 10511 vol*covol 23734 volcvol 23735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-sup 8742 df-inf 8743 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-z 11819 df-uz 12083 df-q 12187 df-rp 12229 df-ioo 12581 df-ico 12583 df-icc 12584 df-fz 12732 df-fl 13000 df-seq 13208 df-exp 13268 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-ovol 23736 df-vol 23737 |
This theorem is referenced by: 0mbl 23811 icombl1 23835 ioombl 23837 ovolioo 23840 uniiccmbl 23862 volivth 23879 mbfeqalem1 23913 itg10a 23982 itg2uba 24015 itgss3 24086 cntnevol 31060 voliunnfl 34413 volsupnfl 34414 cnambfre 34417 snmbl 41743 |
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