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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 34612 | . 2 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | atllat 34587 | . 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 1990 Latclat 17045 AtLatcal 34551 CvLatclc 34552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-atl 34585 df-cvlat 34609 |
This theorem is referenced by: cvlposN 34614 cvlexch2 34616 cvlexchb1 34617 cvlexchb2 34618 cvlatexchb2 34622 cvlatexch1 34623 cvlatexch2 34624 cvlatexch3 34625 cvlcvr1 34626 cvlcvrp 34627 cvlatcvr2 34629 cvlsupr2 34630 cvlsupr7 34635 cvlsupr8 34636 |
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