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 23298 | . . . . . 6 ⊢ ran (,) ∈ TopBases | |
2 | bastg 21504 | . . . . . 6 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | retop 23299 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | sssigagen 31304 | . . . . . 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 3975 | . . . 4 ⊢ ran (,) ⊆ (sigaGen‘(topGen‘ran (,))) |
8 | df-brsiga 31341 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
9 | 7, 8 | sseqtrri 4003 | . . 3 ⊢ ran (,) ⊆ 𝔅ℝ |
10 | eqid 2821 | . . . . 5 ⊢ vol = vol | |
11 | dmvlsiga 31288 | . . . . . . 7 ⊢ dom vol ∈ (sigAlgebra‘ℝ) | |
12 | elrnsiga 31285 | . . . . . . 7 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) → dom vol ∈ ∪ ran sigAlgebra) | |
13 | 11, 12 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → dom vol ∈ ∪ ran sigAlgebra) |
14 | brsigarn 31343 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
15 | elrnsiga 31285 | . . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
16 | 14, 15 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
17 | 13, 16 | ismbfm 31410 | . . . . 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 497 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → ∀𝑥 ∈ 𝔅ℝ (◡𝐹 “ 𝑥) ∈ dom vol) |
20 | ssralv 4032 | . . 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 498 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol)) |
23 | elmapi 8418 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑m ℝ) → 𝐹:ℝ⟶ℝ) | |
24 | unibrsiga 31345 | . . . . 5 ⊢ ∪ 𝔅ℝ = ℝ | |
25 | unidmvol 24071 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
26 | 24, 25 | oveq12i 7157 | . . . 4 ⊢ (∪ 𝔅ℝ ↑m ∪ dom vol) = (ℝ ↑m ℝ) |
27 | 23, 26 | eleq2s 2931 | . . 3 ⊢ (𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol) → 𝐹:ℝ⟶ℝ) |
28 | ismbf 24158 | . . 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 258 | 1 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ⊆ wss 3935 ∪ cuni 4832 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 ↑m cmap 8396 ℝcr 10525 (,)cioo 12728 topGenctg 16701 Topctop 21431 TopBasesctb 21483 volcvol 23993 MblFncmbf 24144 sigAlgebracsiga 31267 sigaGencsigagen 31297 𝔅ℝcbrsiga 31340 MblFnMcmbfm 31408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-xadd 12498 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-rlim 14836 df-sum 15033 df-topgen 16707 df-xmet 20468 df-met 20469 df-top 21432 df-bases 21484 df-ovol 23994 df-vol 23995 df-mbf 24149 df-siga 31268 df-sigagen 31298 df-brsiga 31341 df-mbfm 31409 |
This theorem is referenced by: (None) |
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