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Theorem plelttr 14155
Description: Transitive law for chained less-than-or-equal and less-than. (sspsstr 3315 analog.) (Contributed by NM, 2-May-2012.)
Hypotheses
Ref Expression
pltletr.b  |-  B  =  ( Base `  K
)
pltletr.l  |-  .<_  =  ( le `  K )
pltletr.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
plelttr  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<_  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )

Proof of Theorem plelttr
StepHypRef Expression
1 pltletr.b . . . . 5  |-  B  =  ( Base `  K
)
2 pltletr.l . . . . 5  |-  .<_  =  ( le `  K )
3 pltletr.s . . . . 5  |-  .<  =  ( lt `  K )
41, 2, 3pleval2 14148 . . . 4  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
543adant3r3 1162 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  <->  ( X  .<  Y  \/  X  =  Y ) ) )
61, 3plttr 14153 . . . . 5  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
76exp3a 425 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
8 breq1 4063 . . . . . 6  |-  ( X  =  Y  ->  ( X  .<  Z  <->  Y  .<  Z ) )
98biimprd 214 . . . . 5  |-  ( X  =  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) )
109a1i 10 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =  Y  ->  ( Y 
.<  Z  ->  X  .<  Z ) ) )
117, 10jaod 369 . . 3  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  \/  X  =  Y )  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
125, 11sylbid 206 . 2  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  .<_  Y  ->  ( Y  .<  Z  ->  X  .<  Z ) ) )
1312imp3a 420 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:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292   Basecbs 13195   lecple 13262   Posetcpo 14123   ltcplt 14124
This theorem is referenced by:  athgt  29463  1cvratex  29480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-poset 14129  df-plt 14141
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