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Mirrors > Home > MPE Home > Th. List > Mathboxes > atl0cl | Structured version Visualization version GIF version |
Description: An atomic lattice has a zero element. We can use this in place of op0cl 33489 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
atl0cl.b | ⊢ 𝐵 = (Base‘𝐾) |
atl0cl.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
atl0cl | ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atl0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | atl0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 16864 | . 2 ⊢ (𝐾 ∈ AtLat → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ AtLat) | |
6 | eqid 2610 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | atl0dm 33607 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom (glb‘𝐾)) |
8 | 1, 2, 5, 7 | glbcl 16821 | . 2 ⊢ (𝐾 ∈ AtLat → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
9 | 4, 8 | eqeltrd 2688 | 1 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 lubclub 16765 glbcglb 16766 0.cp0 16860 AtLatcal 33569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-glb 16798 df-p0 16862 df-atl 33603 |
This theorem is referenced by: atlle0 33610 atlltn0 33611 isat3 33612 atnle0 33614 atlen0 33615 atcmp 33616 atcvreq0 33619 pmap0 34069 dia0 35359 dih0cnv 35590 |
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