Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > llnnlt | Structured version Visualization version GIF version | ||
| Description: Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.) |
| Ref | Expression |
|---|---|
| llnnlt.s | β’ < = (ltβπΎ) |
| llnnlt.n | β’ π = (LLinesβπΎ) |
| Ref | Expression |
|---|---|
| llnnlt | β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | llnnlt.s | . . . . 5 β’ < = (ltβπΎ) | |
| 2 | 1 | pltirr 18288 | . . . 4 β’ ((πΎ β HL β§ π β π) β Β¬ π < π) |
| 3 | 2 | 3adant3 1133 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| 4 | breq2 5153 | . . . 4 β’ (π = π β (π < π β π < π)) | |
| 5 | 4 | notbid 318 | . . 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 2733 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 8 | 7, 1 | pltle 18286 | . . . 4 β’ ((πΎ β HL β§ π β π β§ π β π) β (π < π β π(leβπΎ)π)) |
| 9 | llnnlt.n | . . . . 5 β’ π = (LLinesβπΎ) | |
| 10 | 7, 9 | llncmp 38393 | . . . 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 2954 | . 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 3029 | 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 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 lecple 17204 ltcplt 18261 HLchlt 38220 LLinesclln 38362 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| 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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 |
| This theorem is referenced by: lplnnle2at 38412 |
| Copyright terms: Public domain | W3C validator |