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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolnelpln | Structured version Visualization version GIF version |
Description: No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.) |
Ref | Expression |
---|---|
lvolnelpln.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
lvolnelpln.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
lvolnelpln | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 35165 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2770 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lvolnelpln.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
4 | 2, 3 | lvolbase 35379 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2770 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17260 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 575 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → 𝑋(le‘𝐾)𝑋) |
8 | lvolnelpln.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
9 | 5, 8, 3 | lvolnlelpln 35386 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1113 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑃 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 133 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1630 ∈ wcel 2144 class class class wbr 4784 ‘cfv 6031 Basecbs 16063 lecple 16155 Latclat 17252 HLchlt 35152 LPlanesclpl 35293 LVolsclvol 35294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-lat 17253 df-clat 17315 df-oposet 34978 df-ol 34980 df-oml 34981 df-covers 35068 df-ats 35069 df-atl 35100 df-cvlat 35124 df-hlat 35153 df-llines 35299 df-lplanes 35300 df-lvols 35301 |
This theorem is referenced by: lplncvrlvol2 35416 lplncvrlvol 35417 |
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