Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > latpos | Structured version Visualization version GIF version |
Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011.) |
Ref | Expression |
---|---|
latpos | ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | eqid 2610 | . . 3 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
4 | 1, 2, 3 | islat 16870 | . 2 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom (join‘𝐾) = ((Base‘𝐾) × (Base‘𝐾)) ∧ dom (meet‘𝐾) = ((Base‘𝐾) × (Base‘𝐾))))) |
5 | 4 | simplbi 475 | 1 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 × cxp 5036 dom cdm 5038 ‘cfv 5804 Basecbs 15695 Posetcpo 16763 joincjn 16767 meetcmee 16768 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-lat 16869 |
This theorem is referenced by: latref 16876 latasymb 16877 lattr 16879 latjcom 16882 latjle12 16885 latleeqj1 16886 latmcom 16898 latlem12 16901 latleeqm1 16902 atlpos 33606 cvlposN 33632 hlpos 33670 |
Copyright terms: Public domain | W3C validator |