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 23363 | . . . . . 6 ⊢ ran (,) ∈ TopBases | |
2 | bastg 21568 | . . . . . 6 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | retop 23364 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | sssigagen 31399 | . . . . . 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 31436 | . . . 4 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
9 | 7, 8 | sseqtrri 4003 | . . 3 ⊢ ran (,) ⊆ 𝔅ℝ |
10 | eqid 2821 | . . . . 5 ⊢ vol = vol | |
11 | dmvlsiga 31383 | . . . . . . 7 ⊢ dom vol ∈ (sigAlgebra‘ℝ) | |
12 | elrnsiga 31380 | . . . . . . 7 ⊢ (dom vol ∈ (sigAlgebra‘ℝ) → dom vol ∈ ∪ ran sigAlgebra) | |
13 | 11, 12 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → dom vol ∈ ∪ ran sigAlgebra) |
14 | brsigarn 31438 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
15 | elrnsiga 31380 | . . . . . . 7 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
16 | 14, 15 | mp1i 13 | . . . . . 6 ⊢ (vol = vol → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
17 | 13, 16 | ismbfm 31505 | . . . . 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 499 | . . 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 500 | . . 3 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol)) |
23 | elmapi 8422 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑m ℝ) → 𝐹:ℝ⟶ℝ) | |
24 | unibrsiga 31440 | . . . . 5 ⊢ ∪ 𝔅ℝ = ℝ | |
25 | unidmvol 24136 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
26 | 24, 25 | oveq12i 7162 | . . . 4 ⊢ (∪ 𝔅ℝ ↑m ∪ dom vol) = (ℝ ↑m ℝ) |
27 | 23, 26 | eleq2s 2931 | . . 3 ⊢ (𝐹 ∈ (∪ 𝔅ℝ ↑m ∪ dom vol) → 𝐹:ℝ⟶ℝ) |
28 | ismbf 24223 | . . 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 259 | 1 ⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∪ cuni 4831 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℝcr 10530 (,)cioo 12732 topGenctg 16705 Topctop 21495 TopBasesctb 21547 volcvol 24058 MblFncmbf 24209 sigAlgebracsiga 31362 sigaGencsigagen 31392 𝔅ℝcbrsiga 31435 MblFnMcmbfm 31503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 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 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-topgen 16711 df-xmet 20532 df-met 20533 df-top 21496 df-bases 21548 df-ovol 24059 df-vol 24060 df-mbf 24214 df-siga 31363 df-sigagen 31393 df-brsiga 31436 df-mbfm 31504 |
This theorem is referenced by: (None) |
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