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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 5066 | . . 3 ⊢ (dom vol ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ dom vol𝑥 ⊆ ℝ) | |
2 | mblss 24911 | . . 3 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ) | |
3 | 1, 2 | mprgbir 3072 | . 2 ⊢ dom vol ⊆ 𝒫 ℝ |
4 | rembl 24920 | . . 3 ⊢ ℝ ∈ dom vol | |
5 | cmmbl 24914 | . . . 4 ⊢ (𝑥 ∈ dom vol → (ℝ ∖ 𝑥) ∈ dom vol) | |
6 | 5 | rgen 3067 | . . 3 ⊢ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol |
7 | nnenom 13892 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
8 | 7 | ensymi 8951 | . . . . . . . 8 ⊢ ω ≈ ℕ |
9 | domentr 8960 | . . . . . . . 8 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ) | |
10 | 8, 9 | mpan2 690 | . . . . . . 7 ⊢ (𝑥 ≼ ω → 𝑥 ≼ ℕ) |
11 | elpwi 4572 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 dom vol → 𝑥 ⊆ dom vol) | |
12 | dfss3 3937 | . . . . . . . 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 24937 | . . . . . . 7 ⊢ ((𝑥 ≼ ℕ ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) | |
15 | 10, 13, 14 | syl2anr 598 | . . . . . 6 ⊢ ((𝑥 ∈ 𝒫 dom vol ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
16 | 15 | ex 414 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol)) |
17 | uniiun 5023 | . . . . . 6 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
18 | 17 | eleq1i 2829 | . . . . 5 ⊢ (∪ 𝑥 ∈ dom vol ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
19 | 16, 18 | syl6ibr 252 | . . . 4 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)) |
20 | 19 | rgen 3067 | . . 3 ⊢ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol) |
21 | 4, 6, 20 | 3pm3.2i 1340 | . 2 ⊢ (ℝ ∈ dom vol ∧ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)) |
22 | reex 11149 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | pwex 5340 | . . . 4 ⊢ 𝒫 ℝ ∈ V |
24 | 23, 3 | ssexi 5284 | . . 3 ⊢ dom vol ∈ V |
25 | issiga 32751 | . . 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 710 | 1 ⊢ dom vol ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ∀wral 3065 Vcvv 3448 ∖ cdif 3912 ⊆ wss 3915 𝒫 cpw 4565 ∪ cuni 4870 ∪ ciun 4959 class class class wbr 5110 dom cdm 5638 ‘cfv 6501 ωcom 7807 ≈ cen 8887 ≼ cdom 8888 ℝcr 11057 ℕcn 12160 volcvol 24843 sigAlgebracsiga 32747 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cc 10378 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-siga 32748 |
This theorem is referenced by: volmeas 32870 mbfmvolf 32906 elmbfmvol2 32907 |
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