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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 4548 | . . 3 ⊢ (dom vol ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ dom vol𝑥 ⊆ ℝ) | |
2 | mblss 23106 | . . 3 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ) | |
3 | 1, 2 | mprgbir 2911 | . 2 ⊢ dom vol ⊆ 𝒫 ℝ |
4 | rembl 23115 | . . 3 ⊢ ℝ ∈ dom vol | |
5 | cmmbl 23109 | . . . 4 ⊢ (𝑥 ∈ dom vol → (ℝ ∖ 𝑥) ∈ dom vol) | |
6 | 5 | rgen 2906 | . . 3 ⊢ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol |
7 | nnenom 12641 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
8 | 7 | ensymi 7892 | . . . . . . . 8 ⊢ ω ≈ ℕ |
9 | domentr 7901 | . . . . . . . 8 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ) | |
10 | 8, 9 | mpan2 703 | . . . . . . 7 ⊢ (𝑥 ≼ ω → 𝑥 ≼ ℕ) |
11 | elpwi 4117 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 dom vol → 𝑥 ⊆ dom vol) | |
12 | dfss3 3558 | . . . . . . . 8 ⊢ (𝑥 ⊆ dom vol ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) | |
13 | 11, 12 | sylib 207 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 dom vol → ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
14 | iunmbl2 23132 | . . . . . . 7 ⊢ ((𝑥 ≼ ℕ ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) | |
15 | 10, 13, 14 | syl2anr 494 | . . . . . 6 ⊢ ((𝑥 ∈ 𝒫 dom vol ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
16 | 15 | ex 449 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol)) |
17 | uniiun 4509 | . . . . . 6 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
18 | 17 | eleq1i 2679 | . . . . 5 ⊢ (∪ 𝑥 ∈ dom vol ↔ ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
19 | 16, 18 | syl6ibr 241 | . . . 4 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)) |
20 | 19 | rgen 2906 | . . 3 ⊢ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol) |
21 | 4, 6, 20 | 3pm3.2i 1232 | . 2 ⊢ (ℝ ∈ dom vol ∧ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ 𝒫 dom vol(𝑥 ≼ ω → ∪ 𝑥 ∈ dom vol)) |
22 | reex 9906 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | pwex 4774 | . . . 4 ⊢ 𝒫 ℝ ∈ V |
24 | 23, 3 | ssexi 4731 | . . 3 ⊢ dom vol ∈ V |
25 | issiga 29501 | . . 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 957 | 1 ⊢ dom vol ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 ℝcr 9814 ℕcn 10897 volcvol 23039 sigAlgebracsiga 29497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 df-siga 29498 |
This theorem is referenced by: volmeas 29621 mbfmvolf 29655 elmbfmvol2 29656 |
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