![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolneatN | Structured version Visualization version GIF version |
Description: No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lvolneat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lvolneat.v | ⊢ 𝑉 = (LVols‘𝐾) |
Ref | Expression |
---|---|
lvolneatN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 35172 | . . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
2 | eqid 2771 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
3 | lvolneat.v | . . . 4 ⊢ 𝑉 = (LVols‘𝐾) | |
4 | 2, 3 | lvolbase 35386 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Base‘𝐾)) |
5 | eqid 2771 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 2, 5 | latref 17261 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋(le‘𝐾)𝑋) |
7 | 1, 4, 6 | syl2an 583 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → 𝑋(le‘𝐾)𝑋) |
8 | lvolneat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 5, 8, 3 | lvolnleat 35391 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋(le‘𝐾)𝑋) |
10 | 9 | 3expia 1114 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝐴 → ¬ 𝑋(le‘𝐾)𝑋)) |
11 | 7, 10 | mt2d 133 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) → ¬ 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 Basecbs 16064 lecple 16156 Latclat 17253 Atomscatm 35072 HLchlt 35159 LVolsclvol 35301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 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 6754 df-ov 6796 df-oprab 6797 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-lat 17254 df-clat 17316 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |