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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 35438 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2825 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | llnneat.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
4 | 2, 3 | llnbase 35584 | . . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2825 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17406 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 591 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋(le‘𝐾)𝑋) |
8 | llnneat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 5, 8, 3 | llnnleat 35588 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1156 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → (𝑋 ∈ 𝐴 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 134 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4873 ‘cfv 6123 Basecbs 16222 lecple 16312 Latclat 17398 Atomscatm 35338 HLchlt 35425 LLinesclln 35566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-proset 17281 df-poset 17299 df-plt 17311 df-glb 17328 df-p0 17392 df-lat 17399 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 df-llines 35573 |
This theorem is referenced by: 2atneat 35590 islln2a 35592 cdleme22b 36416 cdlemh 36892 |
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