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Mirrors > Home > MPE Home > Th. List > ordthmeo | Structured version Visualization version GIF version |
Description: An order isomorphism is a homeomorphism on the respective order topologies. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
ordthmeo.1 | β’ π = dom π |
ordthmeo.2 | β’ π = dom π |
Ref | Expression |
---|---|
ordthmeo | β’ ((π β π β§ π β π β§ πΉ Isom π , π (π, π)) β πΉ β ((ordTopβπ )Homeo(ordTopβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordthmeo.1 | . . 3 β’ π = dom π | |
2 | ordthmeo.2 | . . 3 β’ π = dom π | |
3 | 1, 2 | ordthmeolem 23704 | . 2 β’ ((π β π β§ π β π β§ πΉ Isom π , π (π, π)) β πΉ β ((ordTopβπ ) Cn (ordTopβπ))) |
4 | isocnv 7338 | . . 3 β’ (πΉ Isom π , π (π, π) β β‘πΉ Isom π, π (π, π)) | |
5 | 2, 1 | ordthmeolem 23704 | . . . 4 β’ ((π β π β§ π β π β§ β‘πΉ Isom π, π (π, π)) β β‘πΉ β ((ordTopβπ) Cn (ordTopβπ ))) |
6 | 5 | 3com12 1121 | . . 3 β’ ((π β π β§ π β π β§ β‘πΉ Isom π, π (π, π)) β β‘πΉ β ((ordTopβπ) Cn (ordTopβπ ))) |
7 | 4, 6 | syl3an3 1163 | . 2 β’ ((π β π β§ π β π β§ πΉ Isom π , π (π, π)) β β‘πΉ β ((ordTopβπ) Cn (ordTopβπ ))) |
8 | ishmeo 23662 | . 2 β’ (πΉ β ((ordTopβπ )Homeo(ordTopβπ)) β (πΉ β ((ordTopβπ ) Cn (ordTopβπ)) β§ β‘πΉ β ((ordTopβπ) Cn (ordTopβπ )))) | |
9 | 3, 7, 8 | sylanbrc 582 | 1 β’ ((π β π β§ π β π β§ πΉ Isom π , π (π, π)) β πΉ β ((ordTopβπ )Homeo(ordTopβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 β‘ccnv 5677 dom cdm 5678 βcfv 6548 Isom wiso 6549 (class class class)co 7420 ordTopcordt 17480 Cn ccn 23127 Homeochmeo 23656 |
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-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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-ral 3059 df-rex 3068 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-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-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-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-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-fin 8967 df-fi 9434 df-topgen 17424 df-ordt 17482 df-top 22795 df-topon 22812 df-bases 22848 df-cn 23130 df-hmeo 23658 |
This theorem is referenced by: icopnfhmeo 24867 iccpnfhmeo 24869 xrhmeo 24870 xrge0iifhmeo 33537 |
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