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 18296 | . . . 4 β’ ((πΎ β HL β§ π β π) β Β¬ π < π) |
| 3 | 2 | 3adant3 1129 | . . 3 β’ ((πΎ β HL β§ π β π β§ π β π) β Β¬ π < π) |
| 4 | breq2 5143 | . . . 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 2724 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 8 | 7, 1 | pltle 18294 | . . . 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 38897 | . . . 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 2945 | . 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 3020 | 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 1084 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 lecple 17209 ltcplt 18269 HLchlt 38724 LLinesclln 38866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-lat 18393 df-clat 18460 df-oposet 38550 df-ol 38552 df-oml 38553 df-covers 38640 df-ats 38641 df-atl 38672 df-cvlat 38696 df-hlat 38725 df-llines 38873 |
| This theorem is referenced by: lplnnle2at 38916 |
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