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Theorem plttr 14432
Description: The less-than relation is transitive. (psstr 3453 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltnlt.b  |-  B  =  ( Base `  K
)
pltnlt.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
plttr  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )

Proof of Theorem plttr
StepHypRef Expression
1 eqid 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
2 pltnlt.s . . . . . 6  |-  .<  =  ( lt `  K )
31, 2pltle 14423 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  K
) Y ) )
433adant3r3 1165 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  X ( le `  K ) Y ) )
51, 2pltle 14423 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<  Z  ->  Y
( le `  K
) Z ) )
653adant3r1 1163 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( Y  .<  Z  ->  Y ( le `  K ) Z ) )
7 pltnlt.b . . . . 5  |-  B  =  ( Base `  K
)
87, 1postr 14415 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X ( le `  K ) Y  /\  Y ( le `  K ) Z )  ->  X ( le
`  K ) Z ) )
94, 6, 8syl2and 471 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X
( le `  K
) Z ) )
107, 2pltn2lp 14431 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
11103adant3r3 1165 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
12 breq2 4219 . . . . . . 7  |-  ( X  =  Z  ->  ( Y  .<  X  <->  Y  .<  Z ) )
1312anbi2d 686 . . . . . 6  |-  ( X  =  Z  ->  (
( X  .<  Y  /\  Y  .<  X )  <->  ( X  .<  Y  /\  Y  .<  Z ) ) )
1413notbid 287 . . . . 5  |-  ( X  =  Z  ->  ( -.  ( X  .<  Y  /\  Y  .<  X )  <->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1511, 14syl5ibcom 213 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1615necon2ad 2654 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  =/=  Z ) )
179, 16jcad 521 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  ( X ( le `  K ) Z  /\  X  =/=  Z ) ) )
181, 2pltval 14422 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X
( le `  K
) Z  /\  X  =/=  Z ) ) )
19183adant3r2 1164 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X ( le `  K ) Z  /\  X  =/= 
Z ) ) )
2017, 19sylibrd 227 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:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457   Basecbs 13474   lecple 13541   Posetcpo 14402   ltcplt 14403
This theorem is referenced by:  pltletr  14433  plelttr  14434  pospo  14435  ofldchr  24249  hlhgt2  30260  hl0lt1N  30261  lhp0lt  30874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-poset 14408  df-plt 14420
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