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Theorem plttr 14104
Description: The less-than relation is transitive. (psstr 3280 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 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
2 pltnlt.s . . . . . 6  |-  .<  =  ( lt `  K )
31, 2pltle 14095 . . . . 5  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  K
) Y ) )
433adant3r3 1162 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  X ( le `  K ) Y ) )
51, 2pltle 14095 . . . . 5  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .<  Z  ->  Y
( le `  K
) Z ) )
653adant3r1 1160 . . . 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 14087 . . . 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 469 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X
( le `  K
) Z ) )
107, 2pltn2lp 14103 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
11103adant3r3 1162 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
12 breq2 4027 . . . . . . 7  |-  ( X  =  Z  ->  ( Y  .<  X  <->  Y  .<  Z ) )
1312anbi2d 684 . . . . . 6  |-  ( X  =  Z  ->  (
( X  .<  Y  /\  Y  .<  X )  <->  ( X  .<  Y  /\  Y  .<  Z ) ) )
1413notbid 285 . . . . 5  |-  ( X  =  Z  ->  ( -.  ( X  .<  Y  /\  Y  .<  X )  <->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1511, 14syl5ibcom 211 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Z  ->  -.  ( X  .<  Y  /\  Y  .<  Z ) ) )
1615necon2ad 2494 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  =/=  Z ) )
179, 16jcad 519 . 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 14094 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<  Z  <->  ( X
( le `  K
) Z  /\  X  =/=  Z ) ) )
19183adant3r2 1161 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Z  <->  ( X ( le `  K ) Z  /\  X  =/= 
Z ) ) )
2017, 19sylibrd 225 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075
This theorem is referenced by:  pltletr  14105  plelttr  14106  pospo  14107  hlhgt2  29578  hl0lt1N  29579  lhp0lt  30192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-poset 14080  df-plt 14092
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