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Mathbox for Norm Megill |
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| 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 18292 | . . . 4 β’ ((πΎ β HL β§ π β π) β Β¬ π < π) |
| 3 | 2 | 3adant3 1130 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| 4 | breq2 5151 | . . . 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 2730 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 8 | 7, 1 | pltle 18290 | . . . 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 38736 | . . . 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 2951 | . 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 3026 | 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 1085 = wceq 1539 β wcel 2104 class class class wbr 5147 βcfv 6542 lecple 17208 ltcplt 18265 HLchlt 38523 LPlanesclpl 38666 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 |
| This theorem is referenced by: lvolnle3at 38756 |
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