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Mirrors > Home > MPE Home > Th. List > ordttop | Structured version Visualization version GIF version |
Description: The order topology is a topology. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ordttop | ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | ordttopon 23183 | . 2 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅)) |
3 | topontop 22901 | . 2 ⊢ ((ordTop‘𝑅) ∈ (TopOn‘dom 𝑅) → (ordTop‘𝑅) ∈ Top) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 dom cdm 5673 ‘cfv 6544 ordTopcordt 17507 Topctop 22881 TopOnctopon 22898 |
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 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 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-om 7867 df-1o 8486 df-2o 8487 df-en 8965 df-fin 8968 df-fi 9445 df-topgen 17451 df-ordt 17509 df-top 22882 df-topon 22899 df-bases 22935 |
This theorem is referenced by: ordtrest 23192 ordtrest2lem 23193 ordtrest2 23194 ordtt1 23369 ordtrestNEW 33747 ordtrest2NEWlem 33748 ordtrest2NEW 33749 |
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