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Mathbox for Norm Megill |
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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 35338 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 2778 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | atl0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 17427 | . 2 ⊢ (𝐾 ∈ AtLat → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ AtLat) | |
6 | eqid 2778 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | atl0dm 35456 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom (glb‘𝐾)) |
8 | 1, 2, 5, 7 | glbcl 17384 | . 2 ⊢ (𝐾 ∈ AtLat → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
9 | 4, 8 | eqeltrd 2859 | 1 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 Basecbs 16255 lubclub 17328 glbcglb 17329 0.cp0 17423 AtLatcal 35418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-glb 17361 df-p0 17425 df-atl 35452 |
This theorem is referenced by: atlle0 35459 atlltn0 35460 isat3 35461 atnle0 35463 atlen0 35464 atcmp 35465 atcvreq0 35468 pmap0 35919 dia0 37206 dih0cnv 37437 |
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