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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atl0dm | Structured version Visualization version GIF version |
Description: Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.) |
Ref | Expression |
---|---|
atl01dm.b | ⊢ 𝐵 = (Base‘𝐾) |
atl01dm.u | ⊢ 𝑈 = (lub‘𝐾) |
atl01dm.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
atl0dm | ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atl01dm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | atl01dm.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
3 | eqid 2725 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2725 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2725 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 38901 | . 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥))) |
7 | 6 | simp2bi 1143 | 1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 class class class wbr 5149 dom cdm 5678 ‘cfv 6549 Basecbs 17183 lecple 17243 lubclub 18304 glbcglb 18305 0.cp0 18418 Latclat 18426 Atomscatm 38865 AtLatcal 38866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-dm 5688 df-iota 6501 df-fv 6557 df-atl 38900 |
This theorem is referenced by: atl0cl 38905 atl0le 38906 |
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