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Mathbox for Norm Megill |
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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 2777 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2777 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | eqid 2777 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2777 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2777 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 35448 | . 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 1136 | 1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 class class class wbr 4886 dom cdm 5355 ‘cfv 6135 Basecbs 16255 lecple 16345 glbcglb 17329 0.cp0 17423 Latclat 17431 Atomscatm 35412 AtLatcal 35413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-dm 5365 df-iota 6099 df-fv 6143 df-atl 35447 |
This theorem is referenced by: atlpos 35450 atnle 35466 atlatmstc 35468 cvllat 35475 hllat 35512 snatpsubN 35899 |
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