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Theorem pltletr 14105
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3282 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltletr.b  |-  B  =  ( Base `  K
)
pltletr.l  |-  .<_  =  ( le `  K )
pltletr.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltletr  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )

Proof of Theorem pltletr
StepHypRef Expression
1 pltletr.b . . . . . 6  |-  B  =  ( Base `  K
)
2 pltletr.l . . . . . 6  |-  .<_  =  ( le `  K )
3 pltletr.s . . . . . 6  |-  .<  =  ( lt `  K )
41, 2, 3pleval2 14099 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
543adant3r1 1160 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
65adantr 451 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
71, 3plttr 14104 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
87expdimp 426 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<  Z  ->  X  .<  Z ) )
9 breq2 4027 . . . . . 6  |-  ( Y  =  Z  ->  ( X  .<  Y  <->  X  .<  Z ) )
109biimpcd 215 . . . . 5  |-  ( X 
.<  Y  ->  ( Y  =  Z  ->  X  .<  Z ) )
1110adantl 452 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  =  Z  ->  X 
.<  Z ) )
128, 11jaod 369 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  (
( Y  .<  Z  \/  Y  =  Z )  ->  X  .<  Z )
)
136, 12sylbid 206 . 2  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  ->  X  .<  Z ) )
1413expimpd 586 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075
This theorem is referenced by:  cvrletrN  29463  atlen0  29500  atlelt  29627  2atlt  29628  ps-2  29667  llnnleat  29702  lplnnle2at  29730  lvolnle3at  29771  dalemcea  29849  2atm2atN  29974  dia2dimlem2  31255  dia2dimlem3  31256
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-mo 2148  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-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-poset 14080  df-plt 14092
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