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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlrelat1 | Structured version Visualization version GIF version |
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 31134, with ∧ swapped, analog.) (Contributed by NM, 4-Dec-2011.) |
Ref | Expression |
---|---|
hlrelat1.b | ⊢ 𝐵 = (Base‘𝐾) |
hlrelat1.l | ⊢ ≤ = (le‘𝐾) |
hlrelat1.s | ⊢ < = (lt‘𝐾) |
hlrelat1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlrelat1 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlomcmat 37765 | . 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat)) | |
2 | hlrelat1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | hlrelat1.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | hlrelat1.s | . . 3 ⊢ < = (lt‘𝐾) | |
5 | hlrelat1.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 2, 3, 4, 5 | atlrelat1 37721 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
7 | 1, 6 | syl3an1 1163 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 class class class wbr 5103 ‘cfv 6493 Basecbs 17043 lecple 17100 ltcplt 18157 CLatccla 18347 OMLcoml 37575 Atomscatm 37663 AtLatcal 37664 HLchlt 37750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-lat 18281 df-clat 18348 df-oposet 37576 df-ol 37578 df-oml 37579 df-covers 37666 df-ats 37667 df-atl 37698 df-cvlat 37722 df-hlat 37751 |
This theorem is referenced by: hlrelat5N 37802 hlrelat 37803 hl2at 37806 hlrelat3 37813 cvrexchlem 37820 lhpexle3lem 38412 |
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