Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2823 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2823 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2823 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | 1, 2, 3 | isdlat 17805 | . 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 500 | 1 ⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 joincjn 17556 meetcmee 17557 Latclat 17657 DLatcdlat 17803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-dlat 17804 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |