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| Mirrors > Home > MPE Home > Th. List > latnlej | Structured version Visualization version GIF version | ||
| Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.) |
| Ref | Expression |
|---|---|
| latlej.b | β’ π΅ = (BaseβπΎ) |
| latlej.l | β’ β€ = (leβπΎ) |
| latlej.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| latnlej | β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ Β¬ π β€ (π β¨ π)) β (π β π β§ π β π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latlej.b | . . . . . . 7 β’ π΅ = (BaseβπΎ) | |
| 2 | latlej.l | . . . . . . 7 β’ β€ = (leβπΎ) | |
| 3 | latlej.j | . . . . . . 7 β’ β¨ = (joinβπΎ) | |
| 4 | 1, 2, 3 | latlej1 18411 | . . . . . 6 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β€ (π β¨ π)) |
| 5 | 4 | 3adant3r1 1181 | . . . . 5 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β€ (π β¨ π)) |
| 6 | breq1 5151 | . . . . 5 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π))) | |
| 7 | 5, 6 | syl5ibrcom 246 | . . . 4 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π = π β π β€ (π β¨ π))) |
| 8 | 7 | necon3bd 2953 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (Β¬ π β€ (π β¨ π) β π β π)) |
| 9 | 1, 2, 3 | latlej2 18412 | . . . . . 6 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β π β€ (π β¨ π)) |
| 10 | 9 | 3adant3r1 1181 | . . . . 5 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β€ (π β¨ π)) |
| 11 | breq1 5151 | . . . . 5 β’ (π = π β (π β€ (π β¨ π) β π β€ (π β¨ π))) | |
| 12 | 10, 11 | syl5ibrcom 246 | . . . 4 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π = π β π β€ (π β¨ π))) |
| 13 | 12 | necon3bd 2953 | . . 3 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (Β¬ π β€ (π β¨ π) β π β π)) |
| 14 | 8, 13 | jcad 512 | . 2 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (Β¬ π β€ (π β¨ π) β (π β π β§ π β π))) |
| 15 | 14 | 3impia 1116 | 1 β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ Β¬ π β€ (π β¨ π)) β (π β π β§ π β π)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 class class class wbr 5148 βcfv 6543 (class class class)co 7412 Basecbs 17151 lecple 17211 joincjn 18274 Latclat 18394 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-lub 18309 df-join 18311 df-lat 18395 |
| This theorem is referenced by: latnlej1l 18420 latnlej1r 18421 |
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