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Mirrors > Home > MPE Home > Th. List > dlatl | Structured version Visualization version GIF version |
Description: A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
dlatl | ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2798 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | 1, 2, 3 | isdlat 17795 | . 2 ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(meet‘𝐾)(𝑦(join‘𝐾)𝑧)) = ((𝑥(meet‘𝐾)𝑦)(join‘𝐾)(𝑥(meet‘𝐾)𝑧)))) |
5 | 4 | simplbi 501 | 1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 joincjn 17546 meetcmee 17547 Latclat 17647 DLatcdlat 17793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-dlat 17794 |
This theorem is referenced by: (None) |
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