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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 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | eqid 2798 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2798 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2798 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 36595 | . 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 1142 | 1 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 Basecbs 16475 lecple 16564 glbcglb 17545 0.cp0 17639 Latclat 17647 Atomscatm 36559 AtLatcal 36560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-atl 36594 |
This theorem is referenced by: atlpos 36597 atnle 36613 atlatmstc 36615 cvllat 36622 hllat 36659 snatpsubN 37046 |
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