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Theorem pltletr 15458
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3610 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 15452 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
543adant3r1 1205 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
65adantr 465 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
71, 3plttr 15457 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
87expdimp 437 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<  Z  ->  X  .<  Z ) )
9 breq2 4451 . . . . . 6  |-  ( Y  =  Z  ->  ( X  .<  Y  <->  X  .<  Z ) )
109biimpcd 224 . . . . 5  |-  ( X 
.<  Y  ->  ( Y  =  Z  ->  X  .<  Z ) )
1110adantl 466 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  =  Z  ->  X 
.<  Z ) )
128, 11jaod 380 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  (
( Y  .<  Z  \/  Y  =  Z )  ->  X  .<  Z )
)
136, 12sylbid 215 . 2  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  ->  X  .<  Z ) )
1413expimpd 603 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588   Basecbs 14490   lecple 14562   Posetcpo 15427   ltcplt 15428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-poset 15433  df-plt 15445
This theorem is referenced by:  cvrletrN  34088  atlen0  34125  atlelt  34252  2atlt  34253  ps-2  34292  llnnleat  34327  lplnnle2at  34355  lvolnle3at  34396  dalemcea  34474  2atm2atN  34599  dia2dimlem2  35880  dia2dimlem3  35881
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