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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 2734 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2734 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2734 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 39280 | . 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥))) |
7 | 6 | simp2bi 1145 | 1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 class class class wbr 5147 dom cdm 5688 ‘cfv 6562 Basecbs 17244 lecple 17304 lubclub 18366 glbcglb 18367 0.cp0 18480 Latclat 18488 Atomscatm 39244 AtLatcal 39245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-dm 5698 df-iota 6515 df-fv 6570 df-atl 39279 |
This theorem is referenced by: atl0cl 39284 atl0le 39285 |
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