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Theorem plelttr 15801
Description: Transitive law for chained less-than-or-equal and less-than. (sspsstr 3595 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 15794 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
543adant3r3 1205 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
61, 3plttr 15799 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
76expd 434 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
8 breq1 4442 . . . . . 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 378 . . 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 429 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570   Basecbs 14716   lecple 14791   Posetcpo 15768   ltcplt 15769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-preset 15756  df-poset 15774  df-plt 15787
This theorem is referenced by:  isarchi3  27965  archiabllem2c  27973  athgt  35577  1cvratex  35594
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