Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnnlt | Structured version Visualization version GIF version | ||
| Description: Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.) |
| Ref | Expression |
|---|---|
| lplnnlt.s | β’ < = (ltβπΎ) |
| lplnnlt.p | β’ π = (LPlanesβπΎ) |
| Ref | Expression |
|---|---|
| lplnnlt | β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lplnnlt.s | . . . . 5 β’ < = (ltβπΎ) | |
| 2 | 1 | pltirr 18253 | . . . 4 β’ ((πΎ β HL β§ π β π) β Β¬ π < π) |
| 3 | 2 | 3adant3 1132 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| 4 | breq2 5129 | . . . 4 β’ (π = π β (π < π β π < π)) | |
| 5 | 4 | notbid 317 | . . 3 β’ (π = π β (Β¬ π < π β Β¬ π < π)) |
| 6 | 3, 5 | syl5ibcom 244 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π) β (π = π β Β¬ π < π)) |
| 7 | eqid 2731 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 8 | 7, 1 | pltle 18251 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β π) β (π < π β π(leβπΎ)π)) |
| 9 | lplnnlt.p | . . . . 5 β’ π = (LPlanesβπΎ) | |
| 10 | 7, 9 | lplncmp 38131 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β π) β (π(leβπΎ)π β π = π)) |
| 11 | 8, 10 | sylibd 238 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β (π < π β π = π)) |
| 12 | 11 | necon3ad 2952 | . 2 β’ ((πΎ β HL β§ π β π β§ π β π) β (π β π β Β¬ π < π)) |
| 13 | 6, 12 | pm2.61dne 3027 | 1 β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5125 βcfv 6516 lecple 17169 ltcplt 18226 HLchlt 37918 LPlanesclpl 38061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-proset 18213 df-poset 18231 df-plt 18248 df-lub 18264 df-glb 18265 df-join 18266 df-meet 18267 df-p0 18343 df-lat 18350 df-clat 18417 df-oposet 37744 df-ol 37746 df-oml 37747 df-covers 37834 df-ats 37835 df-atl 37866 df-cvlat 37890 df-hlat 37919 df-llines 38067 df-lplanes 38068 |
| This theorem is referenced by: lvolnle3at 38151 |
| Copyright terms: Public domain | W3C validator |