MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lattrd Unicode version

Theorem lattrd 14160
Description: A lattice ordering is transitive. Deduction version of lattr 14158. (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 14158 . . 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 1685   class class class wbr 4024   ` cfv 5221   Basecbs 13144   lecple 13211   Latclat 14147
This theorem is referenced by:  latmlej11  14192  latjass  14197  lubun  14223  lubunNEW  28442  cvlcvr1  28808  exatleN  28872  2atjm  28913  2llnmat  28992  llnmlplnN  29007  2llnjaN  29034  2lplnja  29087  dalem5  29135  lncmp  29251  2lnat  29252  2llnma1b  29254  cdlema1N  29259  paddasslem5  29292  paddasslem12  29299  paddasslem13  29300  dalawlem3  29341  dalawlem5  29343  dalawlem6  29344  dalawlem7  29345  dalawlem8  29346  dalawlem11  29349  dalawlem12  29350  pl42lem1N  29447  lhpexle2lem  29477  lhpexle3lem  29479  4atexlemtlw  29535  4atexlemc  29537  cdleme15  29746  cdleme17b  29755  cdleme22e  29812  cdleme22eALTN  29813  cdleme23a  29817  cdleme28a  29838  cdleme30a  29846  cdleme32e  29913  cdleme35b  29918  trlord  30037  cdlemg10  30109  cdlemg11b  30110  cdlemg17a  30129  cdlemg35  30181  tendococl  30240  tendopltp  30248  cdlemi1  30286  cdlemk11  30317  cdlemk5u  30329  cdlemk11u  30339  cdlemk52  30422  dialss  30515  diaglbN  30524  diaintclN  30527  dia2dimlem1  30533  cdlemm10N  30587  djajN  30606  dibglbN  30635  dibintclN  30636  diblss  30639  cdlemn10  30675  dihord1  30687  dihord2pre2  30695  dihopelvalcpre  30717  dihord5apre  30731  dihmeetlem1N  30759  dihglblem2N  30763  dihmeetlem2N  30768  dihglbcpreN  30769  dihmeetlem3N  30774
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-poset 14076  df-lat 14148
  Copyright terms: Public domain W3C validator