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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 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2610 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2610 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 33604 | . 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 1069 | 1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 Basecbs 15695 lecple 15775 glbcglb 16766 0.cp0 16860 Latclat 16868 Atomscatm 33568 AtLatcal 33569 |
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-ne 2782 df-ral 2901 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-dm 5048 df-iota 5768 df-fv 5812 df-atl 33603 |
This theorem is referenced by: atlpos 33606 atnle 33622 atlatmstc 33624 cvllat 33631 hllat 33668 snatpsubN 34054 |
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