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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 36322 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 2823 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | atl0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 17653 | . 2 ⊢ (𝐾 ∈ AtLat → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ AtLat) | |
6 | eqid 2823 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | atl0dm 36440 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom (glb‘𝐾)) |
8 | 1, 2, 5, 7 | glbcl 17610 | . 2 ⊢ (𝐾 ∈ AtLat → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
9 | 4, 8 | eqeltrd 2915 | 1 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Basecbs 16485 lubclub 17554 glbcglb 17555 0.cp0 17649 AtLatcal 36402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-glb 17587 df-p0 17651 df-atl 36436 |
This theorem is referenced by: atlle0 36443 atlltn0 36444 isat3 36445 atnle0 36447 atlen0 36448 atcmp 36449 atcvreq0 36452 pmap0 36903 dia0 38190 dih0cnv 38421 |
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