![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > orvclteel | Structured version Visualization version GIF version |
Description: Preimage maps produced by the "less than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
dstfrv.1 | β’ (π β π β Prob) |
dstfrv.2 | β’ (π β π β (rRndVarβπ)) |
orvclteel.1 | β’ (π β π΄ β β) |
Ref | Expression |
---|---|
orvclteel | β’ (π β (πβRV/π β€ π΄) β dom π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . 2 β’ (π β π β Prob) | |
2 | dstfrv.2 | . 2 β’ (π β π β (rRndVarβπ)) | |
3 | orvclteel.1 | . 2 β’ (π β π΄ β β) | |
4 | rexr 11290 | . . . . . . . 8 β’ (π₯ β β β π₯ β β*) | |
5 | 4 | ad2antrl 726 | . . . . . . 7 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β π₯ β β*) |
6 | mnflt 13135 | . . . . . . . . 9 β’ (π₯ β β β -β < π₯) | |
7 | 6 | ad2antrl 726 | . . . . . . . 8 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β -β < π₯) |
8 | simprr 771 | . . . . . . . 8 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β π₯ β€ π΄) | |
9 | 7, 8 | jca 510 | . . . . . . 7 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β (-β < π₯ β§ π₯ β€ π΄)) |
10 | 5, 9 | jca 510 | . . . . . 6 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) |
11 | simprl 769 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β β*) | |
12 | 3 | adantr 479 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π΄ β β) |
13 | simprrl 779 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β -β < π₯) | |
14 | simprrr 780 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β€ π΄) | |
15 | xrre 13180 | . . . . . . . 8 β’ (((π₯ β β* β§ π΄ β β) β§ (-β < π₯ β§ π₯ β€ π΄)) β π₯ β β) | |
16 | 11, 12, 13, 14, 15 | syl22anc 837 | . . . . . . 7 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β β) |
17 | 16, 14 | jca 510 | . . . . . 6 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β (π₯ β β β§ π₯ β€ π΄)) |
18 | 10, 17 | impbida 799 | . . . . 5 β’ (π β ((π₯ β β β§ π₯ β€ π΄) β (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄)))) |
19 | 18 | rabbidva2 3421 | . . . 4 β’ (π β {π₯ β β β£ π₯ β€ π΄} = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) |
20 | mnfxr 11301 | . . . . 5 β’ -β β β* | |
21 | 3 | rexrd 11294 | . . . . 5 β’ (π β π΄ β β*) |
22 | iocval 13393 | . . . . 5 β’ ((-β β β* β§ π΄ β β*) β (-β(,]π΄) = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) | |
23 | 20, 21, 22 | sylancr 585 | . . . 4 β’ (π β (-β(,]π΄) = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) |
24 | 19, 23 | eqtr4d 2768 | . . 3 β’ (π β {π₯ β β β£ π₯ β€ π΄} = (-β(,]π΄)) |
25 | iocmnfcld 24703 | . . . 4 β’ (π΄ β β β (-β(,]π΄) β (Clsdβ(topGenβran (,)))) | |
26 | 3, 25 | syl 17 | . . 3 β’ (π β (-β(,]π΄) β (Clsdβ(topGenβran (,)))) |
27 | 24, 26 | eqeltrd 2825 | . 2 β’ (π β {π₯ β β β£ π₯ β€ π΄} β (Clsdβ(topGenβran (,)))) |
28 | 1, 2, 3, 27 | orrvccel 34143 | 1 β’ (π β (πβRV/π β€ π΄) β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3419 class class class wbr 5143 dom cdm 5672 ran crn 5673 βcfv 6543 (class class class)co 7416 βcr 11137 -βcmnf 11276 β*cxr 11277 < clt 11278 β€ cle 11279 (,)cioo 13356 (,]cioc 13357 topGenctg 17418 Clsdccld 22938 Probcprb 34084 rRndVarcrrv 34117 βRV/πcorvc 34132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-ioo 13360 df-ioc 13361 df-topgen 17424 df-top 22814 df-bases 22867 df-cld 22941 df-esum 33704 df-siga 33785 df-sigagen 33815 df-brsiga 33858 df-meas 33872 df-mbfm 33926 df-prob 34085 df-rrv 34118 df-orvc 34133 |
This theorem is referenced by: dstfrvunirn 34151 dstfrvinc 34153 dstfrvclim1 34154 |
Copyright terms: Public domain | W3C validator |