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Theorem pltletr 15140
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3461 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 15134 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<_  Z  <->  ( Y  .<  Z  \/  Y  =  Z ) ) )
543adant3r1 1196 . . . 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 15139 . . . . 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 4295 . . . . . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4291   ` cfv 5417   Basecbs 14173   lecple 14244   Posetcpo 15109   ltcplt 15110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-poset 15115  df-plt 15127
This theorem is referenced by:  cvrletrN  32916  atlen0  32953  atlelt  33080  2atlt  33081  ps-2  33120  llnnleat  33155  lplnnle2at  33183  lvolnle3at  33224  dalemcea  33302  2atm2atN  33427  dia2dimlem2  34708  dia2dimlem3  34709
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