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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfresmf | Structured version Visualization version GIF version |
Description: A real-valued measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
mbfresmf.1 | β’ (π β πΉ β MblFn) |
mbfresmf.2 | β’ (π β ran πΉ β β) |
mbfresmf.3 | β’ π = dom vol |
Ref | Expression |
---|---|
mbfresmf | β’ (π β πΉ β (SMblFnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . 2 β’ β²ππ | |
2 | mbfresmf.3 | . . . 4 β’ π = dom vol | |
3 | 2 | a1i 11 | . . 3 β’ (π β π = dom vol) |
4 | dmvolsal 44673 | . . . 4 β’ dom vol β SAlg | |
5 | 4 | a1i 11 | . . 3 β’ (π β dom vol β SAlg) |
6 | 3, 5 | eqeltrd 2834 | . 2 β’ (π β π β SAlg) |
7 | mbfresmf.1 | . . . 4 β’ (π β πΉ β MblFn) | |
8 | mbfdmssre 44327 | . . . 4 β’ (πΉ β MblFn β dom πΉ β β) | |
9 | 7, 8 | syl 17 | . . 3 β’ (π β dom πΉ β β) |
10 | 2 | unieqi 4879 | . . . 4 β’ βͺ π = βͺ dom vol |
11 | unidmvol 24921 | . . . 4 β’ βͺ dom vol = β | |
12 | 10, 11 | eqtri 2761 | . . 3 β’ βͺ π = β |
13 | 9, 12 | sseqtrrdi 3996 | . 2 β’ (π β dom πΉ β βͺ π) |
14 | mbff 25005 | . . . . 5 β’ (πΉ β MblFn β πΉ:dom πΉβΆβ) | |
15 | ffn 6669 | . . . . 5 β’ (πΉ:dom πΉβΆβ β πΉ Fn dom πΉ) | |
16 | 7, 14, 15 | 3syl 18 | . . . 4 β’ (π β πΉ Fn dom πΉ) |
17 | mbfresmf.2 | . . . 4 β’ (π β ran πΉ β β) | |
18 | 16, 17 | jca 513 | . . 3 β’ (π β (πΉ Fn dom πΉ β§ ran πΉ β β)) |
19 | df-f 6501 | . . 3 β’ (πΉ:dom πΉβΆβ β (πΉ Fn dom πΉ β§ ran πΉ β β)) | |
20 | 18, 19 | sylibr 233 | . 2 β’ (π β πΉ:dom πΉβΆβ) |
21 | 20 | adantr 482 | . . . . 5 β’ ((π β§ π β β) β πΉ:dom πΉβΆβ) |
22 | rexr 11206 | . . . . . 6 β’ (π β β β π β β*) | |
23 | 22 | adantl 483 | . . . . 5 β’ ((π β§ π β β) β π β β*) |
24 | 21, 23 | preimaioomnf 45046 | . . . 4 β’ ((π β§ π β β) β (β‘πΉ β (-β(,)π)) = {π₯ β dom πΉ β£ (πΉβπ₯) < π}) |
25 | 24 | eqcomd 2739 | . . 3 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} = (β‘πΉ β (-β(,)π))) |
26 | 4 | elexi 3463 | . . . . . 6 β’ dom vol β V |
27 | 2, 26 | eqeltri 2830 | . . . . 5 β’ π β V |
28 | 27 | a1i 11 | . . . 4 β’ ((π β§ π β β) β π β V) |
29 | 7 | dmexd 7843 | . . . . 5 β’ (π β dom πΉ β V) |
30 | 29 | adantr 482 | . . . 4 β’ ((π β§ π β β) β dom πΉ β V) |
31 | mbfima 25010 | . . . . . . 7 β’ ((πΉ β MblFn β§ πΉ:dom πΉβΆβ) β (β‘πΉ β (-β(,)π)) β dom vol) | |
32 | 7, 20, 31 | syl2anc 585 | . . . . . 6 β’ (π β (β‘πΉ β (-β(,)π)) β dom vol) |
33 | 32, 3 | eleqtrrd 2837 | . . . . 5 β’ (π β (β‘πΉ β (-β(,)π)) β π) |
34 | 33 | adantr 482 | . . . 4 β’ ((π β§ π β β) β (β‘πΉ β (-β(,)π)) β π) |
35 | cnvimass 6034 | . . . . 5 β’ (β‘πΉ β (-β(,)π)) β dom πΉ | |
36 | dfss 3929 | . . . . . 6 β’ ((β‘πΉ β (-β(,)π)) β dom πΉ β (β‘πΉ β (-β(,)π)) = ((β‘πΉ β (-β(,)π)) β© dom πΉ)) | |
37 | 36 | biimpi 215 | . . . . 5 β’ ((β‘πΉ β (-β(,)π)) β dom πΉ β (β‘πΉ β (-β(,)π)) = ((β‘πΉ β (-β(,)π)) β© dom πΉ)) |
38 | 35, 37 | ax-mp 5 | . . . 4 β’ (β‘πΉ β (-β(,)π)) = ((β‘πΉ β (-β(,)π)) β© dom πΉ) |
39 | 28, 30, 34, 38 | elrestd 43406 | . . 3 β’ ((π β§ π β β) β (β‘πΉ β (-β(,)π)) β (π βΎt dom πΉ)) |
40 | 25, 39 | eqeltrd 2834 | . 2 β’ ((π β§ π β β) β {π₯ β dom πΉ β£ (πΉβπ₯) < π} β (π βΎt dom πΉ)) |
41 | 1, 6, 13, 20, 40 | issmfd 45062 | 1 β’ (π β πΉ β (SMblFnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 β© cin 3910 β wss 3911 βͺ cuni 4866 class class class wbr 5106 β‘ccnv 5633 dom cdm 5634 ran crn 5635 β cima 5637 Fn wfn 6492 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11054 βcr 11055 -βcmnf 11192 β*cxr 11193 < clt 11194 (,)cioo 13270 βΎt crest 17307 volcvol 24843 MblFncmbf 24994 SAlgcsalg 44635 SMblFncsmblfn 45022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cc 10376 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-disj 5072 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xadd 13039 df-ioo 13274 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 df-sum 15577 df-rest 17309 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-mbf 24999 df-salg 44636 df-smblfn 45023 |
This theorem is referenced by: mbfpsssmf 45110 |
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