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Mirrors > Home > MPE Home > Th. List > ordtrestixx | Structured version Visualization version GIF version |
Description: The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
ordtrestixx.1 | β’ π΄ β β* |
ordtrestixx.2 | β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯[,]π¦) β π΄) |
Ref | Expression |
---|---|
ordtrestixx | β’ ((ordTopβ β€ ) βΎt π΄) = (ordTopβ( β€ β© (π΄ Γ π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ledm 18543 | . . . 4 β’ β* = dom β€ | |
2 | letsr 18546 | . . . . 5 β’ β€ β TosetRel | |
3 | 2 | a1i 11 | . . . 4 β’ (β€ β β€ β TosetRel ) |
4 | ordtrestixx.1 | . . . . 5 β’ π΄ β β* | |
5 | 4 | a1i 11 | . . . 4 β’ (β€ β π΄ β β*) |
6 | 4 | sseli 3979 | . . . . . . 7 β’ (π₯ β π΄ β π₯ β β*) |
7 | 4 | sseli 3979 | . . . . . . 7 β’ (π¦ β π΄ β π¦ β β*) |
8 | iccval 13363 | . . . . . . 7 β’ ((π₯ β β* β§ π¦ β β*) β (π₯[,]π¦) = {π§ β β* β£ (π₯ β€ π§ β§ π§ β€ π¦)}) | |
9 | 6, 7, 8 | syl2an 597 | . . . . . 6 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯[,]π¦) = {π§ β β* β£ (π₯ β€ π§ β§ π§ β€ π¦)}) |
10 | ordtrestixx.2 | . . . . . 6 β’ ((π₯ β π΄ β§ π¦ β π΄) β (π₯[,]π¦) β π΄) | |
11 | 9, 10 | eqsstrrd 4022 | . . . . 5 β’ ((π₯ β π΄ β§ π¦ β π΄) β {π§ β β* β£ (π₯ β€ π§ β§ π§ β€ π¦)} β π΄) |
12 | 11 | adantl 483 | . . . 4 β’ ((β€ β§ (π₯ β π΄ β§ π¦ β π΄)) β {π§ β β* β£ (π₯ β€ π§ β§ π§ β€ π¦)} β π΄) |
13 | 1, 3, 5, 12 | ordtrest2 22708 | . . 3 β’ (β€ β (ordTopβ( β€ β© (π΄ Γ π΄))) = ((ordTopβ β€ ) βΎt π΄)) |
14 | 13 | eqcomd 2739 | . 2 β’ (β€ β ((ordTopβ β€ ) βΎt π΄) = (ordTopβ( β€ β© (π΄ Γ π΄)))) |
15 | 14 | mptru 1549 | 1 β’ ((ordTopβ β€ ) βΎt π΄) = (ordTopβ( β€ β© (π΄ Γ π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β€wtru 1543 β wcel 2107 {crab 3433 β© cin 3948 β wss 3949 class class class wbr 5149 Γ cxp 5675 βcfv 6544 (class class class)co 7409 β*cxr 11247 β€ cle 11249 [,]cicc 13327 βΎt crest 17366 ordTopcordt 17445 TosetRel ctsr 18518 |
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-pre-lttri 11184 ax-pre-lttrn 11185 |
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-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-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-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-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-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-icc 13331 df-rest 17368 df-topgen 17389 df-ordt 17447 df-ps 18519 df-tsr 18520 df-top 22396 df-topon 22413 df-bases 22449 |
This theorem is referenced by: ordtresticc 22727 icopnfhmeo 24459 |
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