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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 23175 | . 2 β’ ((π β π β§ π β π β§ πΉ Isom π , π (π, π)) β πΉ β ((ordTopβπ ) Cn (ordTopβπ))) |
4 | isocnv 7279 | . . 3 β’ (πΉ Isom π , π (π, π) β β‘πΉ Isom π, π (π, π)) | |
5 | 2, 1 | ordthmeolem 23175 | . . . 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 1124 | . . 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 1166 | . 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 23133 | . 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 584 | 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 1088 = wceq 1542 β wcel 2107 β‘ccnv 5636 dom cdm 5637 βcfv 6500 Isom wiso 6501 (class class class)co 7361 ordTopcordt 17389 Cn ccn 22598 Homeochmeo 23127 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-fin 8893 df-fi 9355 df-topgen 17333 df-ordt 17391 df-top 22266 df-topon 22283 df-bases 22319 df-cn 22601 df-hmeo 23129 |
This theorem is referenced by: icopnfhmeo 24329 iccpnfhmeo 24331 xrhmeo 24332 xrge0iifhmeo 32581 |
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