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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 | β’ (πΉ β (dom volMblFnMπ β) β πΉ β MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retopbas 24498 | . . . . . 6 β’ ran (,) β TopBases | |
2 | bastg 22690 | . . . . . 6 β’ (ran (,) β TopBases β ran (,) β (topGenβran (,))) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ ran (,) β (topGenβran (,)) |
4 | retop 24499 | . . . . . 6 β’ (topGenβran (,)) β Top | |
5 | sssigagen 33438 | . . . . . 6 β’ ((topGenβran (,)) β Top β (topGenβran (,)) β (sigaGenβ(topGenβran (,)))) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 β’ (topGenβran (,)) β (sigaGenβ(topGenβran (,))) |
7 | 3, 6 | sstri 3992 | . . . 4 β’ ran (,) β (sigaGenβ(topGenβran (,))) |
8 | df-brsiga 33475 | . . . 4 β’ π β = (sigaGenβ(topGenβran (,))) | |
9 | 7, 8 | sseqtrri 4020 | . . 3 β’ ran (,) β π β |
10 | eqid 2731 | . . . . 5 β’ vol = vol | |
11 | dmvlsiga 33422 | . . . . . . 7 β’ dom vol β (sigAlgebraββ) | |
12 | elrnsiga 33419 | . . . . . . 7 β’ (dom vol β (sigAlgebraββ) β dom vol β βͺ ran sigAlgebra) | |
13 | 11, 12 | mp1i 13 | . . . . . 6 β’ (vol = vol β dom vol β βͺ ran sigAlgebra) |
14 | brsigarn 33477 | . . . . . . 7 β’ π β β (sigAlgebraββ) | |
15 | elrnsiga 33419 | . . . . . . 7 β’ (π β β (sigAlgebraββ) β π β β βͺ ran sigAlgebra) | |
16 | 14, 15 | mp1i 13 | . . . . . 6 β’ (vol = vol β π β β βͺ ran sigAlgebra) |
17 | 13, 16 | ismbfm 33544 | . . . . 5 β’ (vol = vol β (πΉ β (dom volMblFnMπ β) β (πΉ β (βͺ π β βm βͺ dom vol) β§ βπ₯ β π β (β‘πΉ β π₯) β dom vol))) |
18 | 10, 17 | ax-mp 5 | . . . 4 β’ (πΉ β (dom volMblFnMπ β) β (πΉ β (βͺ π β βm βͺ dom vol) β§ βπ₯ β π β (β‘πΉ β π₯) β dom vol)) |
19 | 18 | simprbi 496 | . . 3 β’ (πΉ β (dom volMblFnMπ β) β βπ₯ β π β (β‘πΉ β π₯) β dom vol) |
20 | ssralv 4051 | . . 3 β’ (ran (,) β π β β (βπ₯ β π β (β‘πΉ β π₯) β dom vol β βπ₯ β ran (,)(β‘πΉ β π₯) β dom vol)) | |
21 | 9, 19, 20 | mpsyl 68 | . 2 β’ (πΉ β (dom volMblFnMπ β) β βπ₯ β ran (,)(β‘πΉ β π₯) β dom vol) |
22 | 18 | simplbi 497 | . . 3 β’ (πΉ β (dom volMblFnMπ β) β πΉ β (βͺ π β βm βͺ dom vol)) |
23 | elmapi 8846 | . . . 4 β’ (πΉ β (β βm β) β πΉ:ββΆβ) | |
24 | unibrsiga 33479 | . . . . 5 β’ βͺ π β = β | |
25 | unidmvol 25291 | . . . . 5 β’ βͺ dom vol = β | |
26 | 24, 25 | oveq12i 7424 | . . . 4 β’ (βͺ π β βm βͺ dom vol) = (β βm β) |
27 | 23, 26 | eleq2s 2850 | . . 3 β’ (πΉ β (βͺ π β βm βͺ dom vol) β πΉ:ββΆβ) |
28 | ismbf 25378 | . . 3 β’ (πΉ:ββΆβ β (πΉ β MblFn β βπ₯ β ran (,)(β‘πΉ β π₯) β dom vol)) | |
29 | 22, 27, 28 | 3syl 18 | . 2 β’ (πΉ β (dom volMblFnMπ β) β (πΉ β MblFn β βπ₯ β ran (,)(β‘πΉ β π₯) β dom vol)) |
30 | 21, 29 | mpbird 256 | 1 β’ (πΉ β (dom volMblFnMπ β) β πΉ β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 βwral 3060 β wss 3949 βͺ cuni 4909 β‘ccnv 5676 dom cdm 5677 ran crn 5678 β cima 5680 βΆwf 6540 βcfv 6544 (class class class)co 7412 βm cmap 8823 βcr 11112 (,)cioo 13329 topGenctg 17388 Topctop 22616 TopBasesctb 22669 volcvol 25213 MblFncmbf 25364 sigAlgebracsiga 33401 sigaGencsigagen 33431 π βcbrsiga 33474 MblFnMcmbfm 33542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cc 10433 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-oi 9508 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-xadd 13098 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-topgen 17394 df-xmet 21138 df-met 21139 df-top 22617 df-bases 22670 df-ovol 25214 df-vol 25215 df-mbf 25369 df-siga 33402 df-sigagen 33432 df-brsiga 33475 df-mbfm 33543 |
This theorem is referenced by: (None) |
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