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Theorem pltletr 16168
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3577 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 16162 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
543adant3r1 1214 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
65adantr 466 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
71, 3plttr 16167 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
87expdimp 438 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<  Z  ->  X  .<  Z ) )
9 breq2 4430 . . . . . 6  |-  ( Y  =  Z  ->  ( X  .<  Y  <->  X  .<  Z ) )
109biimpcd 227 . . . . 5  |-  ( X 
.<  Y  ->  ( Y  =  Z  ->  X  .<  Z ) )
1110adantl 467 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  =  Z  ->  X 
.<  Z ) )
128, 11jaod 381 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  (
( Y  .<  Z  \/  Y  =  Z )  ->  X  .<  Z )
)
136, 12sylbid 218 . 2  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  ->  X  .<  Z ) )
1413expimpd 606 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601   Basecbs 15084   lecple 15159   Posetcpo 16136   ltcplt 16137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-preset 16124  df-poset 16142  df-plt 16155
This theorem is referenced by:  cvrletrN  32548  atlen0  32585  atlelt  32712  2atlt  32713  ps-2  32752  llnnleat  32787  lplnnle2at  32815  lvolnle3at  32856  dalemcea  32934  2atm2atN  33059  dia2dimlem2  34342  dia2dimlem3  34343
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