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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvclteinc | Structured version Visualization version GIF version |
Description: Preimage maps produced by the "less than or equal to" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.) |
Ref | Expression |
---|---|
dstfrv.1 | β’ (π β π β Prob) |
dstfrv.2 | β’ (π β π β (rRndVarβπ)) |
orvclteinc.1 | β’ (π β π΄ β β) |
orvclteinc.2 | β’ (π β π΅ β β) |
orvclteinc.3 | β’ (π β π΄ β€ π΅) |
Ref | Expression |
---|---|
orvclteinc | β’ (π β (πβRV/π β€ π΄) β (πβRV/π β€ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . . . . 5 β’ (π β π β Prob) | |
2 | dstfrv.2 | . . . . 5 β’ (π β π β (rRndVarβπ)) | |
3 | 1, 2 | rrvf2 33977 | . . . 4 β’ (π β π:dom πβΆβ) |
4 | 3 | ffund 6715 | . . 3 β’ (π β Fun π) |
5 | simp2 1134 | . . . . . 6 β’ ((π β§ π₯ β β β§ π₯ β€ π΄) β π₯ β β) | |
6 | orvclteinc.1 | . . . . . . 7 β’ (π β π΄ β β) | |
7 | 6 | 3ad2ant1 1130 | . . . . . 6 β’ ((π β§ π₯ β β β§ π₯ β€ π΄) β π΄ β β) |
8 | orvclteinc.2 | . . . . . . 7 β’ (π β π΅ β β) | |
9 | 8 | 3ad2ant1 1130 | . . . . . 6 β’ ((π β§ π₯ β β β§ π₯ β€ π΄) β π΅ β β) |
10 | simp3 1135 | . . . . . 6 β’ ((π β§ π₯ β β β§ π₯ β€ π΄) β π₯ β€ π΄) | |
11 | orvclteinc.3 | . . . . . . 7 β’ (π β π΄ β€ π΅) | |
12 | 11 | 3ad2ant1 1130 | . . . . . 6 β’ ((π β§ π₯ β β β§ π₯ β€ π΄) β π΄ β€ π΅) |
13 | 5, 7, 9, 10, 12 | letrd 11375 | . . . . 5 β’ ((π β§ π₯ β β β§ π₯ β€ π΄) β π₯ β€ π΅) |
14 | 13 | 3expia 1118 | . . . 4 β’ ((π β§ π₯ β β) β (π₯ β€ π΄ β π₯ β€ π΅)) |
15 | 14 | ss2rabdv 4068 | . . 3 β’ (π β {π₯ β β β£ π₯ β€ π΄} β {π₯ β β β£ π₯ β€ π΅}) |
16 | sspreima 7063 | . . 3 β’ ((Fun π β§ {π₯ β β β£ π₯ β€ π΄} β {π₯ β β β£ π₯ β€ π΅}) β (β‘π β {π₯ β β β£ π₯ β€ π΄}) β (β‘π β {π₯ β β β£ π₯ β€ π΅})) | |
17 | 4, 15, 16 | syl2anc 583 | . 2 β’ (π β (β‘π β {π₯ β β β£ π₯ β€ π΄}) β (β‘π β {π₯ β β β£ π₯ β€ π΅})) |
18 | 1, 2, 6 | orrvcval4 33993 | . 2 β’ (π β (πβRV/π β€ π΄) = (β‘π β {π₯ β β β£ π₯ β€ π΄})) |
19 | 1, 2, 8 | orrvcval4 33993 | . 2 β’ (π β (πβRV/π β€ π΅) = (β‘π β {π₯ β β β£ π₯ β€ π΅})) |
20 | 17, 18, 19 | 3sstr4d 4024 | 1 β’ (π β (πβRV/π β€ π΄) β (πβRV/π β€ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 β wcel 2098 {crab 3426 β wss 3943 class class class wbr 5141 β‘ccnv 5668 dom cdm 5669 β cima 5672 Fun wfun 6531 βcfv 6537 (class class class)co 7405 βcr 11111 β€ cle 11253 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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
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-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 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-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13334 df-topgen 17398 df-top 22751 df-bases 22804 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: dstfrvinc 34005 dstfrvclim1 34006 |
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