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Mirrors > Home > MPE Home > Th. List > ioorinv2 | Structured version Visualization version GIF version |
Description: The function πΉ is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by AV, 13-Sep-2020.) |
Ref | Expression |
---|---|
ioorf.1 | β’ πΉ = (π₯ β ran (,) β¦ if(π₯ = β , β¨0, 0β©, β¨inf(π₯, β*, < ), sup(π₯, β*, < )β©)) |
Ref | Expression |
---|---|
ioorinv2 | β’ ((π΄(,)π΅) β β β (πΉβ(π΄(,)π΅)) = β¨π΄, π΅β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioorebas 13431 | . . 3 β’ (π΄(,)π΅) β ran (,) | |
2 | ioorf.1 | . . . 4 β’ πΉ = (π₯ β ran (,) β¦ if(π₯ = β , β¨0, 0β©, β¨inf(π₯, β*, < ), sup(π₯, β*, < )β©)) | |
3 | 2 | ioorval 25453 | . . 3 β’ ((π΄(,)π΅) β ran (,) β (πΉβ(π΄(,)π΅)) = if((π΄(,)π΅) = β , β¨0, 0β©, β¨inf((π΄(,)π΅), β*, < ), sup((π΄(,)π΅), β*, < )β©)) |
4 | 1, 3 | ax-mp 5 | . 2 β’ (πΉβ(π΄(,)π΅)) = if((π΄(,)π΅) = β , β¨0, 0β©, β¨inf((π΄(,)π΅), β*, < ), sup((π΄(,)π΅), β*, < )β©) |
5 | ifnefalse 4535 | . . 3 β’ ((π΄(,)π΅) β β β if((π΄(,)π΅) = β , β¨0, 0β©, β¨inf((π΄(,)π΅), β*, < ), sup((π΄(,)π΅), β*, < )β©) = β¨inf((π΄(,)π΅), β*, < ), sup((π΄(,)π΅), β*, < )β©) | |
6 | n0 4341 | . . . . . . 7 β’ ((π΄(,)π΅) β β β βπ₯ π₯ β (π΄(,)π΅)) | |
7 | eliooxr 13385 | . . . . . . . 8 β’ (π₯ β (π΄(,)π΅) β (π΄ β β* β§ π΅ β β*)) | |
8 | 7 | exlimiv 1925 | . . . . . . 7 β’ (βπ₯ π₯ β (π΄(,)π΅) β (π΄ β β* β§ π΅ β β*)) |
9 | 6, 8 | sylbi 216 | . . . . . 6 β’ ((π΄(,)π΅) β β β (π΄ β β* β§ π΅ β β*)) |
10 | 9 | simpld 494 | . . . . 5 β’ ((π΄(,)π΅) β β β π΄ β β*) |
11 | 9 | simprd 495 | . . . . 5 β’ ((π΄(,)π΅) β β β π΅ β β*) |
12 | id 22 | . . . . 5 β’ ((π΄(,)π΅) β β β (π΄(,)π΅) β β ) | |
13 | df-ioo 13331 | . . . . . 6 β’ (,) = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯ < π§ β§ π§ < π¦)}) | |
14 | idd 24 | . . . . . 6 β’ ((π€ β β* β§ π΅ β β*) β (π€ < π΅ β π€ < π΅)) | |
15 | xrltle 13131 | . . . . . 6 β’ ((π€ β β* β§ π΅ β β*) β (π€ < π΅ β π€ β€ π΅)) | |
16 | idd 24 | . . . . . 6 β’ ((π΄ β β* β§ π€ β β*) β (π΄ < π€ β π΄ < π€)) | |
17 | xrltle 13131 | . . . . . 6 β’ ((π΄ β β* β§ π€ β β*) β (π΄ < π€ β π΄ β€ π€)) | |
18 | 13, 14, 15, 16, 17 | ixxlb 13349 | . . . . 5 β’ ((π΄ β β* β§ π΅ β β* β§ (π΄(,)π΅) β β ) β inf((π΄(,)π΅), β*, < ) = π΄) |
19 | 10, 11, 12, 18 | syl3anc 1368 | . . . 4 β’ ((π΄(,)π΅) β β β inf((π΄(,)π΅), β*, < ) = π΄) |
20 | 13, 14, 15, 16, 17 | ixxub 13348 | . . . . 5 β’ ((π΄ β β* β§ π΅ β β* β§ (π΄(,)π΅) β β ) β sup((π΄(,)π΅), β*, < ) = π΅) |
21 | 10, 11, 12, 20 | syl3anc 1368 | . . . 4 β’ ((π΄(,)π΅) β β β sup((π΄(,)π΅), β*, < ) = π΅) |
22 | 19, 21 | opeq12d 4876 | . . 3 β’ ((π΄(,)π΅) β β β β¨inf((π΄(,)π΅), β*, < ), sup((π΄(,)π΅), β*, < )β© = β¨π΄, π΅β©) |
23 | 5, 22 | eqtrd 2766 | . 2 β’ ((π΄(,)π΅) β β β if((π΄(,)π΅) = β , β¨0, 0β©, β¨inf((π΄(,)π΅), β*, < ), sup((π΄(,)π΅), β*, < )β©) = β¨π΄, π΅β©) |
24 | 4, 23 | eqtrid 2778 | 1 β’ ((π΄(,)π΅) β β β (πΉβ(π΄(,)π΅)) = β¨π΄, π΅β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 βwex 1773 β wcel 2098 β wne 2934 β c0 4317 ifcif 4523 β¨cop 4629 class class class wbr 5141 β¦ cmpt 5224 ran crn 5670 βcfv 6536 (class class class)co 7404 supcsup 9434 infcinf 9435 0cc0 11109 β*cxr 11248 < clt 11249 (,)cioo 13327 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 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-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-iun 4992 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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-ioo 13331 |
This theorem is referenced by: ioorinv 25455 ioorcl 25456 |
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