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 37063 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2736 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | llnneat.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
4 | 2, 3 | llnbase 37209 | . . 3 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2736 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17901 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 599 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋(le‘𝐾)𝑋) |
8 | llnneat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 5, 8, 3 | llnnleat 37213 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1123 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → (𝑋 ∈ 𝐴 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 138 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 Basecbs 16666 lecple 16756 Latclat 17891 Atomscatm 36963 HLchlt 37050 LLinesclln 37191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-proset 17756 df-poset 17774 df-plt 17790 df-glb 17807 df-p0 17885 df-lat 17892 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 |
This theorem is referenced by: 2atneat 37215 islln2a 37217 cdleme22b 38041 cdlemh 38517 |
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