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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 2771 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2771 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2771 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 35917 | . 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥))) |
7 | 6 | simp2bi 1127 | 1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∀wral 3081 ∃wrex 3082 class class class wbr 4925 dom cdm 5403 ‘cfv 6185 Basecbs 16337 lecple 16426 lubclub 17422 glbcglb 17423 0.cp0 17517 Latclat 17525 Atomscatm 35881 AtLatcal 35882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-dm 5413 df-iota 6149 df-fv 6193 df-atl 35916 |
This theorem is referenced by: atl0cl 35921 atl0le 35922 |
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