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Theorem lattrd 14164
Description: A lattice ordering is transitive. Deduction version of lattr 14162. (Contributed by NM, 3-Sep-2012.)
Hypotheses
Ref Expression
lattrd.b  |-  B  =  ( Base `  K
)
lattrd.l  |-  .<_  =  ( le `  K )
lattrd.1  |-  ( ph  ->  K  e.  Lat )
lattrd.2  |-  ( ph  ->  X  e.  B )
lattrd.3  |-  ( ph  ->  Y  e.  B )
lattrd.4  |-  ( ph  ->  Z  e.  B )
lattrd.5  |-  ( ph  ->  X  .<_  Y )
lattrd.6  |-  ( ph  ->  Y  .<_  Z )
Assertion
Ref Expression
lattrd  |-  ( ph  ->  X  .<_  Z )

Proof of Theorem lattrd
StepHypRef Expression
1 lattrd.5 . 2  |-  ( ph  ->  X  .<_  Y )
2 lattrd.6 . 2  |-  ( ph  ->  Y  .<_  Z )
3 lattrd.1 . . 3  |-  ( ph  ->  K  e.  Lat )
4 lattrd.2 . . 3  |-  ( ph  ->  X  e.  B )
5 lattrd.3 . . 3  |-  ( ph  ->  Y  e.  B )
6 lattrd.4 . . 3  |-  ( ph  ->  Z  e.  B )
7 lattrd.b . . . 4  |-  B  =  ( Base `  K
)
8 lattrd.l . . . 4  |-  .<_  =  ( le `  K )
97, 8lattr 14162 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
103, 4, 5, 6, 9syl13anc 1184 . 2  |-  ( ph  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
111, 2, 10mp2and 660 1  |-  ( ph  ->  X  .<_  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Latclat 14151
This theorem is referenced by:  latmlej11  14196  latjass  14201  lubun  14227  lubunNEW  29163  cvlcvr1  29529  exatleN  29593  2atjm  29634  2llnmat  29713  llnmlplnN  29728  2llnjaN  29755  2lplnja  29808  dalem5  29856  lncmp  29972  2lnat  29973  2llnma1b  29975  cdlema1N  29980  paddasslem5  30013  paddasslem12  30020  paddasslem13  30021  dalawlem3  30062  dalawlem5  30064  dalawlem6  30065  dalawlem7  30066  dalawlem8  30067  dalawlem11  30070  dalawlem12  30071  pl42lem1N  30168  lhpexle2lem  30198  lhpexle3lem  30200  4atexlemtlw  30256  4atexlemc  30258  cdleme15  30467  cdleme17b  30476  cdleme22e  30533  cdleme22eALTN  30534  cdleme23a  30538  cdleme28a  30559  cdleme30a  30567  cdleme32e  30634  cdleme35b  30639  trlord  30758  cdlemg10  30830  cdlemg11b  30831  cdlemg17a  30850  cdlemg35  30902  tendococl  30961  tendopltp  30969  cdlemi1  31007  cdlemk11  31038  cdlemk5u  31050  cdlemk11u  31060  cdlemk52  31143  dialss  31236  diaglbN  31245  diaintclN  31248  dia2dimlem1  31254  cdlemm10N  31308  djajN  31327  dibglbN  31356  dibintclN  31357  diblss  31360  cdlemn10  31396  dihord1  31408  dihord2pre2  31416  dihopelvalcpre  31438  dihord5apre  31452  dihmeetlem1N  31480  dihglblem2N  31484  dihmeetlem2N  31489  dihglbcpreN  31490  dihmeetlem3N  31495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-poset 14080  df-lat 14152
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