Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > latnlej2r | Structured version Visualization version GIF version |
Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.) |
Ref | Expression |
---|---|
latlej.b | β’ π΅ = (BaseβπΎ) |
latlej.l | β’ β€ = (leβπΎ) |
latlej.j | β’ β¨ = (joinβπΎ) |
Ref | Expression |
---|---|
latnlej2r | β’ ((πΎ β Lat β§ (π β π΅ β§ π β π΅ β§ π β π΅) β§ Β¬ π β€ (π β¨ π)) β Β¬ π β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | latlej.l | . . 3 β’ β€ = (leβπΎ) | |
3 | latlej.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | 1, 2, 3 | latnlej2 18274 | . 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 (Β¬ π β€ π β§ Β¬ π β€ π)) |
5 | 4 | simprd 496 | 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 Β¬ π β€ π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 class class class wbr 5092 βcfv 6479 (class class class)co 7337 Basecbs 17009 lecple 17066 joincjn 18126 Latclat 18246 |
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 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-poset 18128 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-lat 18247 |
This theorem is referenced by: 4noncolr3 37721 |
Copyright terms: Public domain | W3C validator |