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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnneat | Structured version Visualization version GIF version |
Description: A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.) |
Ref | Expression |
---|---|
llnneat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
llnneat.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
llnneat | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38222 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2733 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | llnneat.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
4 | 2, 3 | llnbase 38369 | . . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2733 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 18391 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 597 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋(le‘𝐾)𝑋) |
8 | llnneat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 5, 8, 3 | llnnleat 38373 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1122 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → (𝑋 ∈ 𝐴 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 136 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5148 ‘cfv 6541 Basecbs 17141 lecple 17201 Latclat 18381 Atomscatm 38122 HLchlt 38209 LLinesclln 38351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-proset 18245 df-poset 18263 df-plt 18280 df-glb 18297 df-p0 18375 df-lat 18382 df-covers 38125 df-ats 38126 df-atl 38157 df-cvlat 38181 df-hlat 38210 df-llines 38358 |
This theorem is referenced by: 2atneat 38375 islln2a 38377 cdleme22b 39201 cdlemh 39677 |
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