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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 11207 | . . . . . 6 ⊢ ℝ ∈ V | |
2 | 1 | pwex 5378 | . . . . 5 ⊢ 𝒫 ℝ ∈ V |
3 | dmvolss 45163 | . . . . 5 ⊢ dom vol ⊆ 𝒫 ℝ | |
4 | 2, 3 | ssexi 5322 | . . . 4 ⊢ dom vol ∈ V |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → dom vol ∈ V) |
6 | 0mbl 25389 | . . . 4 ⊢ ∅ ∈ dom vol | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ∅ ∈ dom vol) |
8 | unidmvol 25391 | . . . 4 ⊢ ∪ dom vol = ℝ | |
9 | 8 | eqcomi 2740 | . . 3 ⊢ ℝ = ∪ dom vol |
10 | cmmbl 25384 | . . . 4 ⊢ (𝑦 ∈ dom vol → (ℝ ∖ 𝑦) ∈ dom vol) | |
11 | 10 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ dom vol) → (ℝ ∖ 𝑦) ∈ dom vol) |
12 | ffvelcdm 7083 | . . . . . 6 ⊢ ((𝑒:ℕ⟶dom vol ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ dom vol) | |
13 | 12 | ralrimiva 3145 | . . . . 5 ⊢ (𝑒:ℕ⟶dom vol → ∀𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol) |
14 | iunmbl 25403 | . . . . 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 481 | . . 3 ⊢ ((⊤ ∧ 𝑒:ℕ⟶dom vol) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ dom vol) |
17 | 5, 7, 9, 11, 16 | issalnnd 45523 | . 2 ⊢ (⊤ → dom vol ∈ SAlg) |
18 | 17 | mptru 1547 | 1 ⊢ dom vol ∈ SAlg |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1541 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ∖ cdif 3945 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 ∪ ciun 4997 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 ℝcr 11115 ℕcn 12219 volcvol 25313 SAlgcsalg 45486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cc 10436 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xadd 13100 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-xmet 21227 df-met 21228 df-ovol 25314 df-vol 25315 df-salg 45487 |
This theorem is referenced by: volmea 45652 mbfresmf 45917 smfmbfcex 45938 |
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