MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plttr Unicode version

Theorem plttr 14120
Description: The less-than relation is transitive. (psstr 3293 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 2296 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
2 pltnlt.s . . . . . 6  |-  .<  =  ( lt `  K )
31, 2pltle 14111 . . . . 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 14111 . . . . 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 14103 . . . 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 14119 . . . . . 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 4043 . . . . . . 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 2507 . . 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 14110 . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091
This theorem is referenced by:  pltletr  14121  plelttr  14122  pospo  14123  hlhgt2  30200  hl0lt1N  30201  lhp0lt  30814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-poset 14096  df-plt 14108
  Copyright terms: Public domain W3C validator