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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnneat | Structured version Visualization version GIF version |
Description: No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
lplnneat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lplnneat.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
lplnneat | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 35944 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2778 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lplnneat.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
4 | 2, 3 | lplnbase 36115 | . . 3 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2778 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17524 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 586 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋(le‘𝐾)𝑋) |
8 | lplnneat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 5, 8, 3 | lplnnleat 36123 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1101 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → (𝑋 ∈ 𝐴 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 134 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 class class class wbr 4930 ‘cfv 6190 Basecbs 16342 lecple 16431 Latclat 17516 Atomscatm 35844 HLchlt 35931 LPlanesclpl 36073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-proset 17399 df-poset 17417 df-plt 17429 df-lub 17445 df-glb 17446 df-join 17447 df-meet 17448 df-p0 17510 df-lat 17517 df-clat 17579 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 |
This theorem is referenced by: llncvrlpln 36139 |
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