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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfmvolf | Structured version Visualization version GIF version |
Description: Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.) |
Ref | Expression |
---|---|
mbfmvolf | ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹:ℝ⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmvlsiga 30733 | . . . . . 6 ⊢ dom vol ∈ (sigAlgebra‘ℝ) | |
2 | issgon 30727 | . . . . . 6 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) ↔ (dom vol ∈ ∪ ran sigAlgebra ∧ ℝ = ∪ dom vol)) | |
3 | 1, 2 | mpbi 222 | . . . . 5 ⊢ (dom vol ∈ ∪ ran sigAlgebra ∧ ℝ = ∪ dom vol) |
4 | 3 | simpli 478 | . . . 4 ⊢ dom vol ∈ ∪ ran sigAlgebra |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → dom vol ∈ ∪ ran sigAlgebra) |
6 | brsigarn 30788 | . . . . . 6 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
7 | issgon 30727 | . . . . . 6 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) ↔ (𝔅ℝ ∈ ∪ ran sigAlgebra ∧ ℝ = ∪ 𝔅ℝ)) | |
8 | 6, 7 | mpbi 222 | . . . . 5 ⊢ (𝔅ℝ ∈ ∪ ran sigAlgebra ∧ ℝ = ∪ 𝔅ℝ) |
9 | 8 | simpli 478 | . . . 4 ⊢ 𝔅ℝ ∈ ∪ ran sigAlgebra |
10 | 9 | a1i 11 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
11 | id 22 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ (dom volMblFnM𝔅ℝ)) | |
12 | 5, 10, 11 | mbfmf 30858 | . 2 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹:∪ dom vol⟶∪ 𝔅ℝ) |
13 | 3 | simpri 481 | . . 3 ⊢ ℝ = ∪ dom vol |
14 | 8 | simpri 481 | . . 3 ⊢ ℝ = ∪ 𝔅ℝ |
15 | 13, 14 | feq23i 6276 | . 2 ⊢ (𝐹:ℝ⟶ℝ ↔ 𝐹:∪ dom vol⟶∪ 𝔅ℝ) |
16 | 12, 15 | sylibr 226 | 1 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹:ℝ⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∪ cuni 4660 dom cdm 5346 ran crn 5347 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 volcvol 23636 sigAlgebracsiga 30711 𝔅ℝcbrsiga 30785 MblFnMcmbfm 30853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cc 9579 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-disj 4844 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-xadd 12240 df-ioo 12474 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-rlim 14604 df-sum 14801 df-topgen 16464 df-xmet 20106 df-met 20107 df-bases 21128 df-ovol 23637 df-vol 23638 df-siga 30712 df-sigagen 30743 df-brsiga 30786 df-mbfm 30854 |
This theorem is referenced by: (None) |
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