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Theorem pltletr 14429
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3454 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 14423 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
543adant3r1 1163 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
65adantr 453 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
71, 3plttr 14428 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
87expdimp 428 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<  Z  ->  X  .<  Z ) )
9 breq2 4217 . . . . . 6  |-  ( Y  =  Z  ->  ( X  .<  Y  <->  X  .<  Z ) )
109biimpcd 217 . . . . 5  |-  ( X 
.<  Y  ->  ( Y  =  Z  ->  X  .<  Z ) )
1110adantl 454 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  =  Z  ->  X 
.<  Z ) )
128, 11jaod 371 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  (
( Y  .<  Z  \/  Y  =  Z )  ->  X  .<  Z )
)
136, 12sylbid 208 . 2  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X  .<  Y )  ->  ( Y  .<_  Z  ->  X  .<  Z ) )
1413expimpd 588 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455   Basecbs 13470   lecple 13537   Posetcpo 14398   ltcplt 14399
This theorem is referenced by:  cvrletrN  30072  atlen0  30109  atlelt  30236  2atlt  30237  ps-2  30276  llnnleat  30311  lplnnle2at  30339  lvolnle3at  30380  dalemcea  30458  2atm2atN  30583  dia2dimlem2  31864  dia2dimlem3  31865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-poset 14404  df-plt 14416
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