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Mirrors > Home > MPE Home > Th. List > Mathboxes > atllat | Structured version Visualization version GIF version |
Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
atllat | ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2818 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | eqid 2818 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2818 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2818 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 36315 | . 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (Base‘𝐾) ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥 ≠ (0.‘𝐾) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝(le‘𝐾)𝑥))) |
7 | 6 | simp1bi 1137 | 1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 Basecbs 16471 lecple 16560 glbcglb 17541 0.cp0 17635 Latclat 17643 Atomscatm 36279 AtLatcal 36280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-dm 5558 df-iota 6307 df-fv 6356 df-atl 36314 |
This theorem is referenced by: atlpos 36317 atnle 36333 atlatmstc 36335 cvllat 36342 hllat 36379 snatpsubN 36766 |
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