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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolnelln | Structured version Visualization version GIF version |
Description: No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
lvolnelln.l | ⊢ 𝑁 = (LLines‘𝐾) |
lvolnelln.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
lvolnelln | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 38171 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2733 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lvolnelln.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
4 | 2, 3 | lvolbase 38387 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2733 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 18390 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 597 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → 𝑋(le‘𝐾)𝑋) |
8 | lvolnelln.l | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
9 | 5, 8, 3 | lvolnlelln 38393 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1122 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑁 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 136 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 Basecbs 17140 lecple 17200 Latclat 18380 HLchlt 38158 LLinesclln 38300 LVolsclvol 38302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-lat 18381 df-clat 18448 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-llines 38307 df-lplanes 38308 df-lvols 38309 |
This theorem is referenced by: lplncvrlvol 38425 |
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