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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 11206 | . . . . . . . 8 β’ (π₯ β β β π₯ β β*) | |
5 | 4 | ad2antrl 727 | . . . . . . 7 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β π₯ β β*) |
6 | mnflt 13049 | . . . . . . . . 9 β’ (π₯ β β β -β < π₯) | |
7 | 6 | ad2antrl 727 | . . . . . . . 8 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β -β < π₯) |
8 | simprr 772 | . . . . . . . 8 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β π₯ β€ π΄) | |
9 | 7, 8 | jca 513 | . . . . . . 7 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β (-β < π₯ β§ π₯ β€ π΄)) |
10 | 5, 9 | jca 513 | . . . . . 6 β’ ((π β§ (π₯ β β β§ π₯ β€ π΄)) β (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) |
11 | simprl 770 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β β*) | |
12 | 3 | adantr 482 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π΄ β β) |
13 | simprrl 780 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β -β < π₯) | |
14 | simprrr 781 | . . . . . . . 8 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β€ π΄) | |
15 | xrre 13094 | . . . . . . . 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 838 | . . . . . . 7 β’ ((π β§ (π₯ β β* β§ (-β < π₯ β§ π₯ β€ π΄))) β π₯ β β) |
17 | 16, 14 | jca 513 | . . . . . 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 800 | . . . . 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 3408 | . . . 4 β’ (π β {π₯ β β β£ π₯ β€ π΄} = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) |
20 | mnfxr 11217 | . . . . 5 β’ -β β β* | |
21 | 3 | rexrd 11210 | . . . . 5 β’ (π β π΄ β β*) |
22 | iocval 13307 | . . . . 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 588 | . . . 4 β’ (π β (-β(,]π΄) = {π₯ β β* β£ (-β < π₯ β§ π₯ β€ π΄)}) |
24 | 19, 23 | eqtr4d 2776 | . . 3 β’ (π β {π₯ β β β£ π₯ β€ π΄} = (-β(,]π΄)) |
25 | iocmnfcld 24148 | . . . 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 2834 | . 2 β’ (π β {π₯ β β β£ π₯ β€ π΄} β (Clsdβ(topGenβran (,)))) |
28 | 1, 2, 3, 27 | orrvccel 33123 | 1 β’ (π β (πβRV/π β€ π΄) β dom π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 class class class wbr 5106 dom cdm 5634 ran crn 5635 βcfv 6497 (class class class)co 7358 βcr 11055 -βcmnf 11192 β*cxr 11193 < clt 11194 β€ cle 11195 (,)cioo 13270 (,]cioc 13271 topGenctg 17324 Clsdccld 22383 Probcprb 33064 rRndVarcrrv 33097 βRV/πcorvc 33112 |
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-ac2 10404 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-iin 4958 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-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-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-acn 9883 df-ac 10057 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-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-ioo 13274 df-ioc 13275 df-topgen 17330 df-top 22259 df-bases 22312 df-cld 22386 df-esum 32684 df-siga 32765 df-sigagen 32795 df-brsiga 32838 df-meas 32852 df-mbfm 32906 df-prob 33065 df-rrv 33098 df-orvc 33113 |
This theorem is referenced by: dstfrvunirn 33131 dstfrvinc 33133 dstfrvclim1 33134 |
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