| 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 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 4 | eqid 2737 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
| 5 | eqid 2737 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 6 | 1, 2, 3, 4, 5 | isatl 39300 | . 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥))) |
| 7 | 6 | simp2bi 1147 | 1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 Basecbs 17247 lecple 17304 lubclub 18355 glbcglb 18356 0.cp0 18468 Latclat 18476 Atomscatm 39264 AtLatcal 39265 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-atl 39299 |
| This theorem is referenced by: atl0cl 39304 atl0le 39305 |
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