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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvllat | Structured version Visualization version GIF version |
Description: An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvllat | ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatl 35939 | . 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | atllat 35914 | . 2 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 Latclat 17526 AtLatcal 35878 CvLatclc 35879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-dm 5414 df-iota 6150 df-fv 6194 df-ov 6978 df-atl 35912 df-cvlat 35936 |
This theorem is referenced by: cvlposN 35941 cvlexch2 35943 cvlexchb1 35944 cvlexchb2 35945 cvlatexchb2 35949 cvlatexch1 35950 cvlatexch2 35951 cvlatexch3 35952 cvlcvr1 35953 cvlcvrp 35954 cvlatcvr2 35956 cvlsupr2 35957 cvlsupr7 35962 cvlsupr8 35963 |
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