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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 13468 | . . 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 25523 | . . 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 4544 | . . 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 4350 | . . . . . . 7 β’ ((π΄(,)π΅) β β β βπ₯ π₯ β (π΄(,)π΅)) | |
7 | eliooxr 13422 | . . . . . . . 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 493 | . . . . 5 β’ ((π΄(,)π΅) β β β π΄ β β*) |
11 | 9 | simprd 494 | . . . . 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 13368 | . . . . . 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 13168 | . . . . . 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 13168 | . . . . . 6 β’ ((π΄ β β* β§ π€ β β*) β (π΄ < π€ β π΄ β€ π€)) | |
18 | 13, 14, 15, 16, 17 | ixxlb 13386 | . . . . 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 13385 | . . . . 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 4886 | . . 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 2768 | . 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 2780 | 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 394 = wceq 1533 βwex 1773 β wcel 2098 β wne 2937 β c0 4326 ifcif 4532 β¨cop 4638 class class class wbr 5152 β¦ cmpt 5235 ran crn 5683 βcfv 6553 (class class class)co 7426 supcsup 9471 infcinf 9472 0cc0 11146 β*cxr 11285 < clt 11286 (,)cioo 13364 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-ioo 13368 |
This theorem is referenced by: ioorinv 25525 ioorcl 25526 |
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