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| 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 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | eqid 2737 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 3 | eqid 2737 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
| 4 | 1, 2, 3 | isdlat 18567 | . 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 497 | 1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 joincjn 18357 meetcmee 18358 Latclat 18476 DLatcdlat 18565 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-dlat 18566 | 
| This theorem is referenced by: (None) | 
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