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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 484 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ⊆ ℝ) | |
2 | elpwi 4568 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
3 | inss2 4190 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
4 | ovolssnul 24867 | . . . . . . . . . 10 ⊢ (((𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝑥 ∩ 𝐴)) = 0) | |
5 | 3, 4 | mp3an1 1449 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(𝑥 ∩ 𝐴)) = 0) |
6 | 5 | adantr 482 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) = 0) |
7 | 6 | oveq1d 7373 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = (0 + (vol*‘(𝑥 ∖ 𝐴)))) |
8 | difss 4092 | . . . . . . . . . . 11 ⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥 | |
9 | ovolsscl 24866 | . . . . . . . . . . 11 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) | |
10 | 8, 9 | mp3an1 1449 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) |
11 | 10 | adantl 483 | . . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) |
12 | 11 | recnd 11188 | . . . . . . . 8 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℂ) |
13 | 12 | addid2d 11361 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (0 + (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘(𝑥 ∖ 𝐴))) |
14 | 7, 13 | eqtrd 2773 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = (vol*‘(𝑥 ∖ 𝐴))) |
15 | simprl 770 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → 𝑥 ⊆ ℝ) | |
16 | ovolss 24865 | . . . . . . 7 ⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ≤ (vol*‘𝑥)) | |
17 | 8, 15, 16 | sylancr 588 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ≤ (vol*‘𝑥)) |
18 | 14, 17 | eqbrtrd 5128 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)) |
19 | 18 | expr 458 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
20 | 2, 19 | sylan2 594 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) ∧ 𝑥 ∈ 𝒫 ℝ) → ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
21 | 20 | ralrimiva 3140 | . 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥))) |
22 | ismbl2 24907 | . 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 584 | 1 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → 𝐴 ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∖ cdif 3908 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4561 class class class wbr 5106 dom cdm 5634 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 0cc0 11056 + caddc 11059 ≤ cle 11195 vol*covol 24842 volcvol 24843 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fl 13703 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-ovol 24844 df-vol 24845 |
This theorem is referenced by: 0mbl 24919 icombl1 24943 ioombl 24945 ovolioo 24948 uniiccmbl 24970 volivth 24987 mbfeqalem1 25021 itg10a 25091 itg2uba 25124 itgss3 25195 cntnevol 32884 voliunnfl 36168 volsupnfl 36169 cnambfre 36172 snmbl 44290 |
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