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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 5097 | . . 3 ⊢ (dom vol ⊆ 𝒫 ℝ ↔ ∀𝑥 ∈ dom vol𝑥 ⊆ ℝ) | |
2 | mblss 25415 | . . 3 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ) | |
3 | 1, 2 | mprgbir 3062 | . 2 ⊢ dom vol ⊆ 𝒫 ℝ |
4 | rembl 25424 | . . 3 ⊢ ℝ ∈ dom vol | |
5 | cmmbl 25418 | . . . 4 ⊢ (𝑥 ∈ dom vol → (ℝ ∖ 𝑥) ∈ dom vol) | |
6 | 5 | rgen 3057 | . . 3 ⊢ ∀𝑥 ∈ dom vol(ℝ ∖ 𝑥) ∈ dom vol |
7 | nnenom 13951 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
8 | 7 | ensymi 9002 | . . . . . . . 8 ⊢ ω ≈ ℕ |
9 | domentr 9011 | . . . . . . . 8 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ) | |
10 | 8, 9 | mpan2 688 | . . . . . . 7 ⊢ (𝑥 ≼ ω → 𝑥 ≼ ℕ) |
11 | elpwi 4604 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 dom vol → 𝑥 ⊆ dom vol) | |
12 | dfss3 3965 | . . . . . . . 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 25441 | . . . . . . 7 ⊢ ((𝑥 ≼ ℕ ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ dom vol) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) | |
15 | 10, 13, 14 | syl2anr 596 | . . . . . 6 ⊢ ((𝑥 ∈ 𝒫 dom vol ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol) |
16 | 15 | ex 412 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 dom vol → (𝑥 ≼ ω → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol)) |
17 | uniiun 5054 | . . . . . 6 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 | |
18 | 17 | eleq1i 2818 | . . . . 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 3057 | . . 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 11203 | . . . . 5 ⊢ ℝ ∈ V | |
23 | 22 | pwex 5371 | . . . 4 ⊢ 𝒫 ℝ ∈ V |
24 | 23, 3 | ssexi 5315 | . . 3 ⊢ dom vol ∈ V |
25 | issiga 33640 | . . 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 708 | 1 ⊢ dom vol ∈ (sigAlgebra‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 𝒫 cpw 4597 ∪ cuni 4902 ∪ ciun 4990 class class class wbr 5141 dom cdm 5669 ‘cfv 6537 ωcom 7852 ≈ cen 8938 ≼ cdom 8939 ℝcr 11111 ℕcn 12216 volcvol 25347 sigAlgebracsiga 33636 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xadd 13099 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 df-xmet 21233 df-met 21234 df-ovol 25348 df-vol 25349 df-siga 33637 |
This theorem is referenced by: volmeas 33759 mbfmvolf 33795 elmbfmvol2 33796 |
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