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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 2738 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | ordttopon 22342 | . 2 ⊢ (𝑅 ∈ 𝑉 → (ordTop‘𝑅) ∈ (TopOn‘dom 𝑅)) |
3 | topontop 22060 | . 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 2106 dom cdm 5591 ‘cfv 6435 ordTopcordt 17208 Topctop 22040 TopOnctopon 22057 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-om 7713 df-1o 8295 df-er 8496 df-en 8732 df-fin 8735 df-fi 9168 df-topgen 17152 df-ordt 17210 df-top 22041 df-topon 22058 df-bases 22094 |
This theorem is referenced by: ordtrest 22351 ordtrest2lem 22352 ordtrest2 22353 ordtt1 22528 ordtrestNEW 31868 ordtrest2NEWlem 31869 ordtrest2NEW 31870 |
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