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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 36514 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lvolnelln.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
4 | 2, 3 | lvolbase 36729 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17663 | . . 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 36735 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1117 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑁 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 138 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 Latclat 17655 HLchlt 36501 LLinesclln 36642 LVolsclvol 36644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 |
This theorem is referenced by: lplncvrlvol 36767 |
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