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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 36493 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lplnneat.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
4 | 2, 3 | lplnbase 36664 | . . 3 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17657 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 597 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋(le‘𝐾)𝑋) |
8 | lplnneat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 5, 8, 3 | lplnnleat 36672 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1117 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → (𝑋 ∈ 𝐴 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 138 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 lecple 16566 Latclat 17649 Atomscatm 36393 HLchlt 36480 LPlanesclpl 36622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 |
This theorem is referenced by: llncvrlpln 36688 |
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