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Mirrors > Home > MPE Home > Th. List > lattr | Structured version Visualization version GIF version |
Description: A lattice ordering is transitive. (sstr 3990 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 18395 | . 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 18277 | . 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 580 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 Basecbs 17148 lecple 17208 Posetcpo 18264 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-ext 2703 ax-nul 5306 |
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-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 df-iota 6495 df-fv 6551 df-poset 18270 df-lat 18389 |
This theorem is referenced by: lattrd 18403 latjlej1 18410 latjlej12 18412 latnlej2 18416 latmlem1 18426 latmlem12 18428 clatleglb 18475 lecmtN 38429 hlrelat2 38577 ps-2 38652 dalem3 38838 dalem17 38854 dalem21 38868 dalem25 38872 linepsubN 38926 pmapsub 38942 cdlemblem 38967 pmapjoin 39026 lhpmcvr4N 39200 4atexlemnclw 39244 cdlemd3 39374 cdleme3g 39408 cdleme3h 39409 cdleme7d 39420 cdleme21c 39501 cdleme32b 39616 cdleme35fnpq 39623 cdleme35f 39628 cdleme48bw 39676 cdlemf1 39735 cdlemg2fv2 39774 cdlemg7fvbwN 39781 cdlemg4 39791 cdlemg6c 39794 cdlemg27a 39866 cdlemg33b0 39875 cdlemg33a 39880 cdlemk3 40007 dia2dimlem1 40238 dihord6b 40434 dihord5apre 40436 dihglbcpreN 40474 |
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