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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 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | eqid 2736 | . . 3 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | eqid 2736 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 36999 | . 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ (0.‘𝐾) → ∃𝑦 ∈ (Atoms‘𝐾)𝑦(le‘𝐾)𝑥))) |
7 | 6 | simp2bi 1148 | 1 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 class class class wbr 5039 dom cdm 5536 ‘cfv 6358 Basecbs 16666 lecple 16756 lubclub 17770 glbcglb 17771 0.cp0 17883 Latclat 17891 Atomscatm 36963 AtLatcal 36964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-dm 5546 df-iota 6316 df-fv 6366 df-atl 36998 |
This theorem is referenced by: atl0cl 37003 atl0le 37004 |
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