![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12327 | . . . 4 β’ -1 β β | |
2 | 1re 11214 | . . . 4 β’ 1 β β | |
3 | neg1lt0 12329 | . . . . 5 β’ -1 < 0 | |
4 | 0lt1 11736 | . . . . 5 β’ 0 < 1 | |
5 | 0re 11216 | . . . . . 6 β’ 0 β β | |
6 | 1, 5, 2 | lttri 11340 | . . . . 5 β’ ((-1 < 0 β§ 0 < 1) β -1 < 1) |
7 | 3, 4, 6 | mp2an 691 | . . . 4 β’ -1 < 1 |
8 | eqid 2733 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
9 | eqid 2733 | . . . . 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 24457 | . . . 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 1462 | . . 3 β’ (π₯ β (0[,]1) β¦ ((π₯ Β· 1) + ((1 β π₯) Β· -1))) β (IIHomeo((TopOpenββfld) βΎt (-1[,]1))) |
12 | hmphi 23281 | . . 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 2733 | . . . . 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 2733 | . . . . 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 24462 | . . . 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 487 | . . 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 23281 | . . 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 23287 | . 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 1542 β wcel 2107 ifcif 4529 class class class wbr 5149 β¦ cmpt 5232 βcfv 6544 Isom wiso 6545 (class class class)co 7409 βcr 11109 0cc0 11110 1c1 11111 + caddc 11113 Β· cmul 11115 +βcpnf 11245 β*cxr 11247 < clt 11248 β€ cle 11249 β cmin 11444 -cneg 11445 / cdiv 11871 -πcxne 13089 [,]cicc 13327 βΎt crest 17366 TopOpenctopn 17367 ordTopcordt 17445 βfldccnfld 20944 Homeochmeo 23257 β chmph 23258 IIcii 24391 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-ordt 17447 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-ps 18519 df-tsr 18520 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cn 22731 df-cnp 22732 df-tx 23066 df-hmeo 23259 df-hmph 23260 df-xms 23826 df-ms 23827 df-tms 23828 df-ii 24393 |
This theorem is referenced by: xrcmp 24464 xrconn 24465 |
Copyright terms: Public domain | W3C validator |