Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmbfmvol2 | Structured version Visualization version GIF version |
Description: Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.) |
Ref | Expression |
---|---|
elmbfmvol2 | ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retopbas 23630 | . . . . . 6 ⊢ ran (,) ∈ TopBases | |
2 | bastg 21835 | . . . . . 6 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | retop 23631 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | sssigagen 31797 | . . . . . 6 ⊢ ((topGen‘ran (,)) ∈ Top → (topGen‘ran (,)) ⊆ (sigaGen‘(topGen‘ran (,)))) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ (topGen‘ran (,)) ⊆ (sigaGen‘(topGen‘ran (,))) |
7 | 3, 6 | sstri 3900 | . . . 4 ⊢ ran (,) ⊆ (sigaGen‘(topGen‘ran (,))) |
8 | df-brsiga 31834 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
9 | 7, 8 | sseqtrri 3928 | . . 3 ⊢ ran (,) ⊆ 𝔅ℝ |
10 | eqid 2734 | . . . . 5 ⊢ vol = vol | |
11 | dmvlsiga 31781 | . . . . . . 7 ⊢ dom vol ∈ (sigAlgebra‘ℝ) | |
12 | elrnsiga 31778 | . . . . . . 7 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) → dom vol ∈ ∪ ran sigAlgebra) | |
13 | 11, 12 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → dom vol ∈ ∪ ran sigAlgebra) |
14 | brsigarn 31836 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
15 | elrnsiga 31778 | . . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
16 | 14, 15 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
17 | 13, 16 | ismbfm 31903 | . . . . 5 ⊢ (vol = vol → (𝐹 ∈ (dom volMblFnM𝔅ℝ) ↔ (𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol))) |
18 | 10, 17 | ax-mp 5 | . . . 4 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) ↔ (𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol) ∧ ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol)) |
19 | 18 | simprbi 500 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol) |
20 | ssralv 3957 | . . 3 ⊢ (ran (,) ⊆ 𝔅ℝ → (∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | |
21 | 9, 19, 20 | mpsyl 68 | . 2 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
22 | 18 | simplbi 501 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol)) |
23 | elmapi 8519 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑m ℝ) → 𝐹:ℝ⟶ℝ) | |
24 | unibrsiga 31838 | . . . . 5 ⊢ ∪ 𝔅ℝ = ℝ | |
25 | unidmvol 24410 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
26 | 24, 25 | oveq12i 7214 | . . . 4 ⊢ (∪ 𝔅ℝ ↑m ∪ dom vol) = (ℝ ↑m ℝ) |
27 | 23, 26 | eleq2s 2852 | . . 3 ⊢ (𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol) → 𝐹:ℝ⟶ℝ) |
28 | ismbf 24497 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | |
29 | 22, 27, 28 | 3syl 18 | . 2 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) |
30 | 21, 29 | mpbird 260 | 1 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ⊆ wss 3857 ∪ cuni 4809 ◡ccnv 5539 dom cdm 5540 ran crn 5541 “ cima 5543 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 ℝcr 10711 (,)cioo 12918 topGenctg 16914 Topctop 21762 TopBasesctb 21814 volcvol 24332 MblFncmbf 24483 sigAlgebracsiga 31760 sigaGencsigagen 31790 𝔅ℝcbrsiga 31833 MblFnMcmbfm 31901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cc 10032 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-disj 5009 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-q 12528 df-rp 12570 df-xadd 12688 df-ioo 12922 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-rlim 15033 df-sum 15233 df-topgen 16920 df-xmet 20328 df-met 20329 df-top 21763 df-bases 21815 df-ovol 24333 df-vol 24334 df-mbf 24488 df-siga 31761 df-sigagen 31791 df-brsiga 31834 df-mbfm 31902 |
This theorem is referenced by: (None) |
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