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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 38565 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 2726 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | atl0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 18390 | . 2 ⊢ (𝐾 ∈ AtLat → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ AtLat) | |
6 | eqid 2726 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | atl0dm 38683 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom (glb‘𝐾)) |
8 | 1, 2, 5, 7 | glbcl 18333 | . 2 ⊢ (𝐾 ∈ AtLat → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
9 | 4, 8 | eqeltrd 2827 | 1 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6536 Basecbs 17151 lubclub 18272 glbcglb 18273 0.cp0 18386 AtLatcal 38645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-glb 18310 df-p0 18388 df-atl 38679 |
This theorem is referenced by: atlle0 38686 atlltn0 38687 isat3 38688 atnle0 38690 atlen0 38691 atcmp 38692 atcvreq0 38695 pmap0 39147 dia0 40434 dih0cnv 40665 |
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