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Mirrors > Home > MPE Home > Th. List > xrhmph | Structured version Visualization version GIF version |
Description: The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
xrhmph | β’ II β (ordTopβ β€ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1rr 12358 | . . . 4 β’ -1 β β | |
2 | 1re 11245 | . . . 4 β’ 1 β β | |
3 | neg1lt0 12360 | . . . . 5 β’ -1 < 0 | |
4 | 0lt1 11767 | . . . . 5 β’ 0 < 1 | |
5 | 0re 11247 | . . . . . 6 β’ 0 β β | |
6 | 1, 5, 2 | lttri 11371 | . . . . 5 β’ ((-1 < 0 β§ 0 < 1) β -1 < 1) |
7 | 3, 4, 6 | mp2an 691 | . . . 4 β’ -1 < 1 |
8 | eqid 2728 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
9 | eqid 2728 | . . . . 5 β’ (π₯ β (0[,]1) β¦ ((π₯ Β· 1) + ((1 β π₯) Β· -1))) = (π₯ β (0[,]1) β¦ ((π₯ Β· 1) + ((1 β π₯) Β· -1))) | |
10 | 8, 9 | icchmeo 24878 | . . . 4 β’ ((-1 β β β§ 1 β β β§ -1 < 1) β (π₯ β (0[,]1) β¦ ((π₯ Β· 1) + ((1 β π₯) Β· -1))) β (IIHomeo((TopOpenββfld) βΎt (-1[,]1)))) |
11 | 1, 2, 7, 10 | mp3an 1458 | . . 3 β’ (π₯ β (0[,]1) β¦ ((π₯ Β· 1) + ((1 β π₯) Β· -1))) β (IIHomeo((TopOpenββfld) βΎt (-1[,]1))) |
12 | hmphi 23694 | . . 3 β’ ((π₯ β (0[,]1) β¦ ((π₯ Β· 1) + ((1 β π₯) Β· -1))) β (IIHomeo((TopOpenββfld) βΎt (-1[,]1))) β II β ((TopOpenββfld) βΎt (-1[,]1))) | |
13 | 11, 12 | ax-mp 5 | . 2 β’ II β ((TopOpenββfld) βΎt (-1[,]1)) |
14 | eqid 2728 | . . . . 5 β’ (π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯)))) = (π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯)))) | |
15 | eqid 2728 | . . . . 5 β’ (π¦ β (-1[,]1) β¦ if(0 β€ π¦, ((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))βπ¦), -π((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))β-π¦))) = (π¦ β (-1[,]1) β¦ if(0 β€ π¦, ((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))βπ¦), -π((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))β-π¦))) | |
16 | 14, 15, 8 | xrhmeo 24884 | . . . 4 β’ ((π¦ β (-1[,]1) β¦ if(0 β€ π¦, ((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))βπ¦), -π((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))β-π¦))) Isom < , < ((-1[,]1), β*) β§ (π¦ β (-1[,]1) β¦ if(0 β€ π¦, ((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))βπ¦), -π((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))β-π¦))) β (((TopOpenββfld) βΎt (-1[,]1))Homeo(ordTopβ β€ ))) |
17 | 16 | simpri 485 | . . 3 β’ (π¦ β (-1[,]1) β¦ if(0 β€ π¦, ((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))βπ¦), -π((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))β-π¦))) β (((TopOpenββfld) βΎt (-1[,]1))Homeo(ordTopβ β€ )) |
18 | hmphi 23694 | . . 3 β’ ((π¦ β (-1[,]1) β¦ if(0 β€ π¦, ((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))βπ¦), -π((π₯ β (0[,]1) β¦ if(π₯ = 1, +β, (π₯ / (1 β π₯))))β-π¦))) β (((TopOpenββfld) βΎt (-1[,]1))Homeo(ordTopβ β€ )) β ((TopOpenββfld) βΎt (-1[,]1)) β (ordTopβ β€ )) | |
19 | 17, 18 | ax-mp 5 | . 2 β’ ((TopOpenββfld) βΎt (-1[,]1)) β (ordTopβ β€ ) |
20 | hmphtr 23700 | . 2 β’ ((II β ((TopOpenββfld) βΎt (-1[,]1)) β§ ((TopOpenββfld) βΎt (-1[,]1)) β (ordTopβ β€ )) β II β (ordTopβ β€ )) | |
21 | 13, 19, 20 | mp2an 691 | 1 β’ II β (ordTopβ β€ ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β wcel 2099 ifcif 4529 class class class wbr 5148 β¦ cmpt 5231 βcfv 6548 Isom wiso 6549 (class class class)co 7420 βcr 11138 0cc0 11139 1c1 11140 + caddc 11142 Β· cmul 11144 +βcpnf 11276 β*cxr 11278 < clt 11279 β€ cle 11280 β cmin 11475 -cneg 11476 / cdiv 11902 -πcxne 13122 [,]cicc 13360 βΎt crest 17402 TopOpenctopn 17403 ordTopcordt 17481 βfldccnfld 21279 Homeochmeo 23670 β chmph 23671 IIcii 24808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-ordt 17483 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-ps 18558 df-tsr 18559 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cn 23144 df-cnp 23145 df-tx 23479 df-hmeo 23672 df-hmph 23673 df-xms 24239 df-ms 24240 df-tms 24241 df-ii 24810 |
This theorem is referenced by: xrcmp 24886 xrconn 24887 |
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