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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmvolsal | Structured version Visualization version GIF version |
Description: Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
dmvolsal | ⊢ dom vol ∈ SAlg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 10617 | . . . . . 6 ⊢ ℝ ∈ V | |
2 | 1 | pwex 5246 | . . . . 5 ⊢ 𝒫 ℝ ∈ V |
3 | dmvolss 42627 | . . . . 5 ⊢ dom vol ⊆ 𝒫 ℝ | |
4 | 2, 3 | ssexi 5190 | . . . 4 ⊢ dom vol ∈ V |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → dom vol ∈ V) |
6 | 0mbl 24143 | . . . 4 ⊢ ∅ ∈ dom vol | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ∅ ∈ dom vol) |
8 | unidmvol 24145 | . . . 4 ⊢ ∪ dom vol = ℝ | |
9 | 8 | eqcomi 2807 | . . 3 ⊢ ℝ = ∪ dom vol |
10 | cmmbl 24138 | . . . 4 ⊢ (𝑦 ∈ dom vol → (ℝ ∖ 𝑦) ∈ dom vol) | |
11 | 10 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ dom vol) → (ℝ ∖ 𝑦) ∈ dom vol) |
12 | ffvelrn 6826 | . . . . . 6 ⊢ ((𝑒:ℕ⟶dom vol ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ dom vol) | |
13 | 12 | ralrimiva 3149 | . . . . 5 ⊢ (𝑒:ℕ⟶dom vol → ∀𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol) |
14 | iunmbl 24157 | . . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝑒:ℕ⟶dom vol → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol) |
16 | 15 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑒:ℕ⟶dom vol) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol) |
17 | 5, 7, 9, 11, 16 | issalnnd 42985 | . 2 ⊢ (⊤ → dom vol ∈ SAlg) |
18 | 17 | mptru 1545 | 1 ⊢ dom vol ∈ SAlg |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1539 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ∅c0 4243 𝒫 cpw 4497 ∪ cuni 4800 ∪ ciun 4881 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 ℝcr 10525 ℕcn 11625 volcvol 24067 SAlgcsalg 42950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-salg 42951 |
This theorem is referenced by: volmea 43113 mbfresmf 43373 smfmbfcex 43393 |
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