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Mirrors > Home > MPE Home > Th. List > ordtresticc | Structured version Visualization version GIF version |
Description: The restriction of the less than order to a closed interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
ordtresticc | β’ ((ordTopβ β€ ) βΎt (π΄[,]π΅)) = (ordTopβ( β€ β© ((π΄[,]π΅) Γ (π΄[,]π΅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 13437 | . 2 β’ (π΄[,]π΅) β β* | |
2 | iccss2 13425 | . 2 β’ ((π₯ β (π΄[,]π΅) β§ π¦ β (π΄[,]π΅)) β (π₯[,]π¦) β (π΄[,]π΅)) | |
3 | 1, 2 | ordtrestixx 23142 | 1 β’ ((ordTopβ β€ ) βΎt (π΄[,]π΅)) = (ordTopβ( β€ β© ((π΄[,]π΅) Γ (π΄[,]π΅)))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β© cin 3939 Γ cxp 5670 βcfv 6542 (class class class)co 7415 β€ cle 11277 [,]cicc 13357 βΎt crest 17399 ordTopcordt 17478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-pre-lttri 11210 ax-pre-lttrn 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fi 9432 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-icc 13361 df-rest 17401 df-topgen 17422 df-ordt 17480 df-ps 18555 df-tsr 18556 df-top 22812 df-topon 22829 df-bases 22865 |
This theorem is referenced by: dfii5 24821 iccpnfhmeo 24886 xrhmeo 24887 icccldii 48048 |
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