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Theorem cvrletrN 30009
Description: Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvrletr.b  |-  B  =  ( Base `  K
)
cvrletr.l  |-  .<_  =  ( le `  K )
cvrletr.s  |-  .<  =  ( lt `  K )
cvrletr.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrletrN  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X C Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )

Proof of Theorem cvrletrN
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  K  e.  Poset )
2 simplr1 999 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  X  e.  B )
3 simplr2 1000 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  Y  e.  B )
4 simpr 448 . . . 4  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  X C Y )
5 cvrletr.b . . . . 5  |-  B  =  ( Base `  K
)
6 cvrletr.s . . . . 5  |-  .<  =  ( lt `  K )
7 cvrletr.c . . . . 5  |-  C  =  (  <o  `  K )
85, 6, 7cvrlt 30006 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
91, 2, 3, 4, 8syl31anc 1187 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  X  .<  Y )
10 cvrletr.l . . . . 5  |-  .<_  =  ( le `  K )
115, 10, 6pltletr 14421 . . . 4  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .<  Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
1211adantr 452 . . 3  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  (
( X  .<  Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
139, 12mpand 657 . 2  |-  ( ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  /\  X C Y )  ->  ( Y  .<_  Z  ->  X  .<  Z ) )
1413expimpd 587 1  |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X C Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4205   ` cfv 5447   Basecbs 13462   lecple 13529   Posetcpo 14390   ltcplt 14391    <o ccvr 29998
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-poset 14396  df-plt 14408  df-covers 30002
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