Nearby theorems
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Mathbox for Norm Megill |
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| 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 18327 | . . . 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 5152 | . . . 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 2728 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 8 | 7, 1 | pltle 18325 | . . . 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 38995 | . . . 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 2950 | . 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 3025 | 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 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 lecple 17240 ltcplt 18300 HLchlt 38822 LLinesclln 38964 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 |
| This theorem is referenced by: lplnnle2at 39014 |
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