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| Mirrors > Home > MPE Home > Th. List > latnle | Structured version Visualization version GIF version | ||
| Description: Equivalent expressions for "not less than" in a lattice. (chnle 31022 analog.) (Contributed by NM, 16-Nov-2011.) |
| Ref | Expression |
|---|---|
| latnle.b | β’ π΅ = (BaseβπΎ) |
| latnle.l | β’ β€ = (leβπΎ) |
| latnle.s | β’ < = (ltβπΎ) |
| latnle.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latnle | β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π β€ π β π < (π β¨ π))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latnle.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 2 | latnle.l | . . . 4 β’ β€ = (leβπΎ) | |
| 3 | latnle.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 4 | 1, 2, 3 | latlej1 18405 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β€ (π β¨ π)) |
| 5 | 4 | biantrurd 533 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β (π β¨ π) β (π β€ (π β¨ π) β§ π β (π β¨ π)))) |
| 6 | 1, 2, 3 | latleeqj1 18408 | . . . . . 6 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β (π β¨ π) = π)) |
| 7 | 6 | 3com23 1126 | . . . . 5 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β (π β¨ π) = π)) |
| 8 | eqcom 2739 | . . . . 5 β’ ((π β¨ π) = π β π = (π β¨ π)) | |
| 9 | 7, 8 | bitrdi 286 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β π = (π β¨ π))) |
| 10 | 1, 3 | latjcom 18404 | . . . . 5 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
| 11 | 10 | eqeq2d 2743 | . . . 4 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π = (π β¨ π) β π = (π β¨ π))) |
| 12 | 9, 11 | bitr4d 281 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β€ π β π = (π β¨ π))) |
| 13 | 12 | necon3bbid 2978 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π β€ π β π β (π β¨ π))) |
| 14 | 1, 3 | latjcl 18396 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
| 15 | latnle.s | . . . 4 β’ < = (ltβπΎ) | |
| 16 | 2, 15 | pltval 18289 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ (π β¨ π) β π΅) β (π < (π β¨ π) β (π β€ (π β¨ π) β§ π β (π β¨ π)))) |
| 17 | 14, 16 | syld3an3 1409 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π < (π β¨ π) β (π β€ (π β¨ π) β§ π β (π β¨ π)))) |
| 18 | 5, 13, 17 | 3bitr4d 310 | 1 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (Β¬ π β€ π β π < (π β¨ π))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 class class class wbr 5148 βcfv 6543 (class class class)co 7411 Basecbs 17148 lecple 17208 ltcplt 18265 joincjn 18268 Latclat 18388 |
| 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 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-lat 18389 |
| This theorem is referenced by: cvlcvr1 38512 hlrelat 38576 hlrelat2 38577 cvr2N 38585 cvrexchlem 38593 |
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