![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lattr | Structured version Visualization version GIF version |
Description: A lattice ordering is transitive. (sstr 3991 analog.) (Contributed by NM, 17-Nov-2011.) |
Ref | Expression |
---|---|
latref.b | β’ π΅ = (BaseβπΎ) |
latref.l | β’ β€ = (leβπΎ) |
Ref | Expression |
---|---|
lattr | 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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latpos 18391 | . 2 β’ (πΎ β Lat β πΎ β Poset) | |
2 | latref.b | . . 3 β’ π΅ = (BaseβπΎ) | |
3 | latref.l | . . 3 β’ β€ = (leβπΎ) | |
4 | 2, 3 | postr 18273 | . 2 β’ ((πΎ β Poset β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β€ π β§ π β€ π) β π β€ π)) |
5 | 1, 4 | sylan 581 | 1 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)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 βcfv 6544 Basecbs 17144 lecple 17204 Posetcpo 18260 Latclat 18384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-dm 5687 df-iota 6496 df-fv 6552 df-poset 18266 df-lat 18385 |
This theorem is referenced by: lattrd 18399 latjlej1 18406 latjlej12 18408 latnlej2 18412 latmlem1 18422 latmlem12 18424 clatleglb 18471 lecmtN 38126 hlrelat2 38274 ps-2 38349 dalem3 38535 dalem17 38551 dalem21 38565 dalem25 38569 linepsubN 38623 pmapsub 38639 cdlemblem 38664 pmapjoin 38723 lhpmcvr4N 38897 4atexlemnclw 38941 cdlemd3 39071 cdleme3g 39105 cdleme3h 39106 cdleme7d 39117 cdleme21c 39198 cdleme32b 39313 cdleme35fnpq 39320 cdleme35f 39325 cdleme48bw 39373 cdlemf1 39432 cdlemg2fv2 39471 cdlemg7fvbwN 39478 cdlemg4 39488 cdlemg6c 39491 cdlemg27a 39563 cdlemg33b0 39572 cdlemg33a 39577 cdlemk3 39704 dia2dimlem1 39935 dihord6b 40131 dihord5apre 40133 dihglbcpreN 40171 |
Copyright terms: Public domain | W3C validator |