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Mathbox for Thierry Arnoux |
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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 11264 | . . . . . . . 8 β’ (π₯ β β β π₯ β β*) | |
5 | 4 | ad2antrl 725 | . . . . . . 7 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β π₯ β β*) |
6 | mnflt 13109 | . . . . . . . . 9 β’ (π₯ β β β -β < π₯) | |
7 | 6 | ad2antrl 725 | . . . . . . . 8 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β -β < π₯) |
8 | simprr 770 | . . . . . . . 8 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β π₯ β€ π΄) | |
9 | 7, 8 | jca 511 | . . . . . . 7 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β (-β < π₯ β§ π₯ β€ π΄)) |
10 | 5, 9 | jca 511 | . . . . . 6 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) |
11 | simprl 768 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β β*) | |
12 | 3 | adantr 480 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π΄ β β) |
13 | simprrl 778 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β -β < π₯) | |
14 | simprrr 779 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β€ π΄) | |
15 | xrre 13154 | . . . . . . . 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 836 | . . . . . . 7 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β β) |
17 | 16, 14 | jca 511 | . . . . . 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 798 | . . . . 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 3428 | . . . 4 β’ (π β {π₯ β β β£ π₯ β€ π΄} = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) |
20 | mnfxr 11275 | . . . . 5 β’ -β β β* | |
21 | 3 | rexrd 11268 | . . . . 5 β’ (π β π΄ β β*) |
22 | iocval 13367 | . . . . 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 586 | . . . 4 β’ (π β (-β(,]π΄) = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) |
24 | 19, 23 | eqtr4d 2769 | . . 3 β’ (π β {π₯ β β β£ π₯ β€ π΄} = (-β(,]π΄)) |
25 | iocmnfcld 24640 | . . . 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 2827 | . 2 β’ (π β {π₯ β β β£ π₯ β€ π΄} β (Clsdβ(topGenβran (,)))) |
28 | 1, 2, 3, 27 | orrvccel 33995 | 1 β’ (π β (πβRV/π β€ π΄) β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3426 class class class wbr 5141 dom cdm 5669 ran crn 5670 βcfv 6537 (class class class)co 7405 βcr 11111 -βcmnf 11250 β*cxr 11251 < clt 11252 β€ cle 11253 (,)cioo 13330 (,]cioc 13331 topGenctg 17392 Clsdccld 22875 Probcprb 33936 rRndVarcrrv 33969 βRV/πcorvc 33984 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-ioo 13334 df-ioc 13335 df-topgen 17398 df-top 22751 df-bases 22804 df-cld 22878 df-esum 33556 df-siga 33637 df-sigagen 33667 df-brsiga 33710 df-meas 33724 df-mbfm 33778 df-prob 33937 df-rrv 33970 df-orvc 33985 |
This theorem is referenced by: dstfrvunirn 34003 dstfrvinc 34005 dstfrvclim1 34006 |
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