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Theorem plttr 15460
Description: The less-than relation is transitive. (psstr 3608 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 2467 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
2 pltnlt.s . . . . . 6  |-  .<  =  ( lt `  K )
31, 2pltle 15451 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  K
) Y ) )
433adant3r3 1207 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  X ( le `  K ) Y ) )
51, 2pltle 15451 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<  Z  ->  Y
( le `  K
) Z ) )
653adant3r1 1205 . . . 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 15443 . . . 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 483 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X
( le `  K
) Z ) )
107, 2pltn2lp 15459 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
11103adant3r3 1207 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
12 breq2 4451 . . . . . . 7  |-  ( X  =  Z  ->  ( Y  .<  X  <->  Y  .<  Z ) )
1312anbi2d 703 . . . . . 6  |-  ( X  =  Z  ->  (
( X  .<  Y  /\  Y  .<  X )  <->  ( X  .<  Y  /\  Y  .<  Z ) ) )
1413notbid 294 . . . . 5  |-  ( X  =  Z  ->  ( -.  ( X  .<  Y  /\  Y  .<  X )  <->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1511, 14syl5ibcom 220 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1615necon2ad 2680 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  =/=  Z ) )
179, 16jcad 533 . 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 15450 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X
( le `  K
) Z  /\  X  =/=  Z ) ) )
19183adant3r2 1206 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X ( le `  K ) Z  /\  X  =/= 
Z ) ) )
2017, 19sylibrd 234 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:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588   Basecbs 14493   lecple 14565   Posetcpo 15430   ltcplt 15431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-poset 15436  df-plt 15448
This theorem is referenced by:  pltletr  15461  plelttr  15462  pospo  15463  archiabllem2c  27498  ofldchr  27564  hlhgt2  34402  hl0lt1N  34403  lhp0lt  35016
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