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Theorem plelttr 15163
Description: Transitive law for chained less-than-or-equal and less-than. (sspsstr 3482 analog.) (Contributed by NM, 2-May-2012.)
Hypotheses
Ref Expression
pltletr.b  |-  B  =  ( Base `  K
)
pltletr.l  |-  .<_  =  ( le `  K )
pltletr.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
plelttr  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )

Proof of Theorem plelttr
StepHypRef Expression
1 pltletr.b . . . . 5  |-  B  =  ( Base `  K
)
2 pltletr.l . . . . 5  |-  .<_  =  ( le `  K )
3 pltletr.s . . . . 5  |-  .<  =  ( lt `  K )
41, 2, 3pleval2 15156 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
543adant3r3 1198 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
61, 3plttr 15161 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
76expd 436 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
8 breq1 4316 . . . . . 6  |-  ( X  =  Y  ->  ( X  .<  Z  <->  Y  .<  Z ) )
98biimprd 223 . . . . 5  |-  ( X  =  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) )
109a1i 11 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Y  ->  ( Y 
.<  Z  ->  X  .<  Z ) ) )
117, 10jaod 380 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  \/  X  =  Y )  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
125, 11sylbid 215 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
1312impd 431 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 4313   ` cfv 5439   Basecbs 14195   lecple 14266   Posetcpo 15131   ltcplt 15132
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 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-iota 5402  df-fun 5441  df-fv 5447  df-poset 15137  df-plt 15149
This theorem is referenced by:  isarchi3  26226  archiabllem2c  26234  athgt  33196  1cvratex  33213
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