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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 13263 | . 2 β’ (π΄[,]π΅) β β* | |
2 | iccss2 13251 | . 2 β’ ((π₯ β (π΄[,]π΅) β§ π¦ β (π΄[,]π΅)) β (π₯[,]π¦) β (π΄[,]π΅)) | |
3 | 1, 2 | ordtrestixx 22479 | 1 β’ ((ordTopβ β€ ) βΎt (π΄[,]π΅)) = (ordTopβ( β€ β© ((π΄[,]π΅) Γ (π΄[,]π΅)))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 β© cin 3897 Γ cxp 5618 βcfv 6479 (class class class)co 7337 β€ cle 11111 [,]cicc 13183 βΎt crest 17228 ordTopcordt 17307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-pre-lttri 11046 ax-pre-lttrn 11047 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fi 9268 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-icc 13187 df-rest 17230 df-topgen 17251 df-ordt 17309 df-ps 18381 df-tsr 18382 df-top 22149 df-topon 22166 df-bases 22202 |
This theorem is referenced by: dfii5 24154 iccpnfhmeo 24214 xrhmeo 24215 icccldii 46563 |
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