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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 39268 | . 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | atllat 39243 | . 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 2104 Latclat 18477 AtLatcal 39207 CvLatclc 39208 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-dm 5693 df-iota 6510 df-fv 6566 df-ov 7428 df-atl 39241 df-cvlat 39265 |
This theorem is referenced by: cvlposN 39270 cvlexch2 39272 cvlexchb1 39273 cvlexchb2 39274 cvlatexchb2 39278 cvlatexch1 39279 cvlatexch2 39280 cvlatexch3 39281 cvlcvr1 39282 cvlcvrp 39283 cvlatcvr2 39285 cvlsupr2 39286 cvlsupr7 39291 cvlsupr8 39292 |
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