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Theorem plttr 14419
Description: The less-than relation is transitive. (psstr 3443 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 2435 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
2 pltnlt.s . . . . . 6  |-  .<  =  ( lt `  K )
31, 2pltle 14410 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  K
) Y ) )
433adant3r3 1164 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  X ( le `  K ) Y ) )
51, 2pltle 14410 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<  Z  ->  Y
( le `  K
) Z ) )
653adant3r1 1162 . . . 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 14402 . . . 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 470 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X
( le `  K
) Z ) )
107, 2pltn2lp 14418 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
11103adant3r3 1164 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
12 breq2 4208 . . . . . . 7  |-  ( X  =  Z  ->  ( Y  .<  X  <->  Y  .<  Z ) )
1312anbi2d 685 . . . . . 6  |-  ( X  =  Z  ->  (
( X  .<  Y  /\  Y  .<  X )  <->  ( X  .<  Y  /\  Y  .<  Z ) ) )
1413notbid 286 . . . . 5  |-  ( X  =  Z  ->  ( -.  ( X  .<  Y  /\  Y  .<  X )  <->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1511, 14syl5ibcom 212 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1615necon2ad 2646 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  =/=  Z ) )
179, 16jcad 520 . 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 14409 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X
( le `  K
) Z  /\  X  =/=  Z ) ) )
19183adant3r2 1163 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X ( le `  K ) Z  /\  X  =/= 
Z ) ) )
2017, 19sylibrd 226 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   ltcplt 14390
This theorem is referenced by:  pltletr  14420  plelttr  14421  pospo  14422  ofldchr  24236  hlhgt2  30113  hl0lt1N  30114  lhp0lt  30727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-poset 14395  df-plt 14407
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