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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmvlsiga | Structured version Visualization version GIF version |
Description: Lebesgue-measurable subsets of ℝ form a sigma-algebra. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.) |
Ref | Expression |
---|---|
dmvlsiga | ⊢ dom vol ∈ (sigAlgebra‘ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwssb 5108 | . . 3 ⊢ (dom vol ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ dom vol𝑥 ⊆ ℝ) | |
2 | mblss 25488 | . . 3 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ) | |
3 | 1, 2 | mprgbir 3065 | . 2 ⊢ dom vol ⊆ 𝒫 ℝ |
4 | rembl 25497 | . . 3 ⊢ ℝ ∈ dom vol | |
5 | cmmbl 25491 | . . . 4 ⊢ (𝑥 ∈ dom vol → (ℝ ∖ 𝑥) ∈ dom vol) | |
6 | 5 | rgen 3060 | . . 3 ⊢ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol |
7 | nnenom 13987 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
8 | 7 | ensymi 9033 | . . . . . . . 8 ⊢ ω ≈ ℕ |
9 | domentr 9042 | . . . . . . . 8 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ) | |
10 | 8, 9 | mpan2 689 | . . . . . . 7 ⊢ (𝑥 ≼ ω → 𝑥 ≼ ℕ) |
11 | elpwi 4613 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 dom vol → 𝑥 ⊆ dom vol) | |
12 | dfss3 3970 | . . . . . . . 8 ⊢ (𝑥 ⊆ dom vol ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) | |
13 | 11, 12 | sylib 217 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 dom vol → ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
14 | iunmbl2 25514 | . . . . . . 7 ⊢ ((𝑥 ≼ ℕ ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) | |
15 | 10, 13, 14 | syl2anr 595 | . . . . . 6 ⊢ ((𝑥 ∈ 𝒫 dom vol ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
16 | 15 | ex 411 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol)) |
17 | uniiun 5065 | . . . . . 6 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
18 | 17 | eleq1i 2820 | . . . . 5 ⊢ (∪ 𝑥 ∈ dom vol ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
19 | 16, 18 | imbitrrdi 251 | . . . 4 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)) |
20 | 19 | rgen 3060 | . . 3 ⊢ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol) |
21 | 4, 6, 20 | 3pm3.2i 1336 | . 2 ⊢ (ℝ ∈ dom vol ∧ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)) |
22 | reex 11239 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | pwex 5384 | . . . 4 ⊢ 𝒫 ℝ ∈ V |
24 | 23, 3 | ssexi 5326 | . . 3 ⊢ dom vol ∈ V |
25 | issiga 33772 | . . 3 ⊢ (dom vol ∈ V → (dom vol ∈ (sigAlgebra‘ℝ) ↔ (dom vol ⊆ 𝒫 ℝ ∧ (ℝ ∈ dom vol ∧ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol))))) | |
26 | 24, 25 | ax-mp 5 | . 2 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) ↔ (dom vol ⊆ 𝒫 ℝ ∧ (ℝ ∈ dom vol ∧ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)))) |
27 | 3, 21, 26 | mpbir2an 709 | 1 ⊢ dom vol ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 𝒫 cpw 4606 ∪ cuni 4912 ∪ ciun 5000 class class class wbr 5152 dom cdm 5682 ‘cfv 6553 ωcom 7878 ≈ cen 8969 ≼ cdom 8970 ℝcr 11147 ℕcn 12252 volcvol 25420 sigAlgebracsiga 33768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cc 10468 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8855 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-oi 9543 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-q 12973 df-rp 13017 df-xadd 13135 df-ioo 13370 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-rlim 15475 df-sum 15675 df-xmet 21286 df-met 21287 df-ovol 25421 df-vol 25422 df-siga 33769 |
This theorem is referenced by: volmeas 33891 mbfmvolf 33927 elmbfmvol2 33928 |
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